''minimum polynomial of a matrix''
...l|characteristic polynomial]] of $A$ and, more generally, it divides every polynomial $f$ such that $f(A)=0$.
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See [[Characteristic polynomial|Characteristic polynomial]].
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...so [[Eigen value|Eigen value]]; [[Characteristic polynomial|Characteristic polynomial]]).
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...xactly, an extension $L$ of a field $K$ is called the splitting field of a polynomial $f$ over the field $K$ if $f$ decomposes over $L$ into linear factors:
...tsc,a_n)$ (see [[Extension of a field]]). A splitting field exists for any polynomial $f\in K[x]$, and it is defined uniquely up to an isomorphism that is the id
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...ots in Jones' construction of his polynomial invariant of links, the Jones polynomial, and Drinfel'd's work on quantum groups (cf. also [[Quantum groups|Quantum
For references, see [[Kauffman polynomial|Kauffman polynomial]]; [[Knot and link diagrams|Knot and link diagrams]].
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The polynomial
The values of the Taylor polynomial and of its derivatives up to order $n$ inclusive at the point $x=x_0$ coinc
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...)c(g_2)$. In particular, a product of primitive polynomials is a primitive polynomial.
...he theory of finite, or [[Galois field]]s, a ''primitive polynomial'' is a polynomial $f$ over a finite field $F$ whose roots are primitive elements, in the sens
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#REDIRECT [[Linearised polynomial]]
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#REDIRECT [[Hilbert polynomial]]
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...s. This theorem can also be used to find the number of negative roots of a polynomial $f(x)$ by considering $f(-x)$.
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#REDIRECT [[Alexander-Conway polynomial]]
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#REDIRECT [[Elementary symmetric polynomial]]
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#REDIRECT [[Jones-Conway polynomial]]
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#REDIRECT [[Jones-Conway polynomial]]
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...t–Lickorish–Millett–Ho polynomial]] and the [[Kauffman polynomial|Kauffman polynomial]]:
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...the basis $v_1,\dots,v_n$. If here $C$ is an infinite integral domain, the polynomial $F$ is defined uniquely.
The polynomial functions on a module $V$ form an associative-commutative $C$-algebra $P(V)
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A trigonometric polynomial of the form
or a similar polynomial in sines. Fejér polynomials are used in constructing continuous functions
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...[[characteristic polynomial]] and [[Minimal polynomial of a matrix|minimal polynomial]] coincide (up to a factor $\pm1$). Equivalently, for each of its distinct
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...ts an element $\alpha \in K$ such that the [[ring of integers]] $O_K$ is a polynomial ring $\mathbb{Z}[\alpha]$. The powers of such a element $\alpha$ constitut
...lynomial|discriminant]] of the [[Minimal polynomial (field theory)|minimal polynomial]] of $\alpha$.
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It is a polynomial of two variables associated to homotopy classes of links in $\mathbf{R}^3$,
...f the graph associated to $D$ (cf. also [[Graph colouring]]). The homotopy polynomial can be generalized to homotopy skein modules of three-dimensional manifolds
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...blishes relations between the roots and the coefficients of a [[Polynomial|polynomial]]. Let $ f( x) $
be a polynomial of degree $ n $
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''additive polynomial''
A [[polynomial]] over a [[field]] of [[Characteristic of a field|characteristic]] $p \ne 0
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...polynomials $\sigma_k(x_1,\ldots,x_n)$ for $k=0,\ldots,n$ where the $k$-th polynomial is obtained by summing all distinct [[monomial]]s which are products of $k$
...] $S_n$, so that any symmetric polynomial in the $x_i$ can be written as a polynomial in the $\sigma_k$.
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...]]], [[#References|[a2]]] and generalized by L.H. Kauffman (the [[Kauffman polynomial]]; cf. also [[Link]]).
...> <TD valign="top"> R.D. Brandt, W.B.R. Lickorish, K.C. Millett, "A polynomial invariant for unoriented knots and links" ''Invent. Math.'' , '''84''' (1
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...s knot]]) are Neuwirth knots. So is every alternating knot whose Alexander polynomial has leading coefficient $\pm1$.
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The normalized version of the Alexander polynomial (cf. also [[Alexander invariants]]). It satisfies the Conway skein relation
...[Knot theory]]). For $z=\sqrt t-1/\sqrt t$ one gets the original Alexander polynomial (defined only up to $\pm t^i$).
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..., $P_M(n) = h_M(n) = \dim_K M_n$. This polynomial is called the ''Hilbert polynomial''.
...e Hilbert polynomial of the ring $R$ is also the name given to the Hilbert polynomial of the projective variety $X$ with respect to the imbedding $X \subset \mat
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$#C+1 = 32 : ~/encyclopedia/old_files/data/C021/C.0201720 Characteristic polynomial
The polynomial
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...r or polynomial or element of a ring; see [[Divisor (of an integer or of a polynomial)]];
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#REDIRECT [[Divisor (of an integer or of a polynomial)]]
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...f a function $f$ in the form of a [[Trigonometric polynomial|trigonometric polynomial]]
...unction at $2n+1$ preassigned points $x_k$ in the interval $[0,2\pi)$. The polynomial has the form
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For references, see [[Kauffman polynomial|Kauffman polynomial]].
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...nt: 1) $A$ is semi-simple; 2) the [[Minimal polynomial of a matrix|minimal polynomial]] of $A$ has no multiple factors in $F[X]$; and 3) the algebra $F[A]$ is a
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A necessary and sufficient condition for all the roots of a polynomial
...art criterion]], and methods for determining the number of real roots of a polynomial are also known.
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...e $\pm1$. '''Littlewood's problem''' asks how large the values of such a polynomial must be on the [[unit circle]] in the [[complex plane]]. The answer to th
A polynomial
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...utely irreducible polynomials of arbitrarily high degree, for example, any polynomial of the form $f(x_1,\ldots,x_{n-1})+x_n$ is absolutely irreducible.
...any irreducible polynomial in a single variable is of degree 1 or 2 and a polynomial of degree 2 is irreducible if and only if its discriminant is negative. Ove
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$#C+1 = 12 : ~/encyclopedia/old_files/data/K110/K.1100100 Kharitonov polynomial theory
...rned with the root locations for a family of polynomials (cf. [[Polynomial|Polynomial]]). A good general reference for this area is [[#References|[a1]]]. The mot
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$#C+1 = 7 : ~/encyclopedia/old_files/data/T094/T.0904230 Trigonometric polynomial,
is called the order of the trigonometric polynomial (provided $ | a _ {n} | + | b _ {n} | > 0 $).
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Every non-trivial knot has a non-trivial [[Jones polynomial]].
...es not hold, as M.B. Thistlethwaite found a $15$-crossing link whose Jones polynomial coincides with a trivial link of two components, cf. Fig.a1. This and simil
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A divisor of a polynomial $A(x)$ is a polynomial $B(x)$ that divides $A(x)$ without remainder (cf. [[Division|Division]]).
...e of $a$, then $a$ is ''irreducible''. For polynomials, see [[Irreducible polynomial]]; for integers, the traditional terminology is [[prime number]].
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''polynomial regression''
exist). The regression is called parabolic (polynomial) if the components of the vector $ {\mathsf E} \{ Y \mid X \} = f( x)
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...omial'' (also, companion or auxiliary polynomial) of the recurrence is the polynomial
It is the characteristic polynomial of the left shift operator acting on the space of all sequences. If $\alph
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...e polynomials one can calculate an approximation of two roots of the given
polynomial. An advantage of the method is that it uses real arithmetic only.
be a given
polynomial with real coefficients and <img align="absmiddle" border="0" src="https://w
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...le of topology has ultimately been reduced to the single assumption that a polynomial of odd degree with real coefficients has a real root.
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''Bernstein form, Bézier polynomial''
The Bernstein polynomial of order $n$ for a function $f$, defined on the closed interval $[0,1]$, is
4 KB (598 words) - 16:55, 1 July 2020
''(algebraic) polynomial''
is called the degree of the polynomial; the polynomial $ P ( z) \equiv 0 $
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A field $k$ is algebraically closed if any polynomial of non-zero degree over $k$ has at
field $k$ each polynomial of degree $n$ over $k$ has exactly $n$ roots
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...g $w_1=w_2=w$, then the resulting polynomial is called the simple matching polynomial of $G$.
...e been investigated [[#References|[a8]]]. The analytical properties of the polynomial have also been investigated [[#References|[a10]]].
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...perfect matching exists. (It should be noted that this is not the [[Tutte polynomial]] of $G$.)
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* A [[divisor (of an integer or of a polynomial)]]
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...gebraic number]]. An algebraic irrationality is the root of an irreducible polynomial of a degree at least two, with rational coefficients.
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be a [[Polynomial|polynomial]] of degree $ \geq 1 $
is irreducible (cf. [[Irreducible polynomial|Irreducible polynomial]]) and, trivially, that the leading coefficient is positive. Are these cond
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$#C+1 = 17 : ~/encyclopedia/old_files/data/A011/A.0101620 Algebraic polynomial of best approximation
A polynomial deviating least from a given function. More precisely, let a measurable fun
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A method for calculating the roots of a polynomial
...mials of degree 3. The parabola method allows one to find all zeros of the polynomial without preliminary information about initial approximations. The convergen
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...ferential equations; it is an analogue of the [[Hilbert polynomial|Hilbert polynomial]].
There exists (see [[#References|[2]]]) a polynomial whose value at points $ s \in \mathbf Z $
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A form in four variables, that is, a homogeneous polynomial (cf. [[Homogeneous function|Homogeneous function]]) in four unknowns with c
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A conjecture on the asymptotic behaviour of a polynomial satisfying the Bunyakovskii condition (cf. also [[Bunyakovskii conjecture|B
be polynomials (cf. [[Polynomial|Polynomial]]) with integer coefficients, of degrees $ d _ {1} \dots d _ {r} \geq 1 $
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''polynomial deviating least from zero''
An algebraic polynomial of degree $n$, with leading coefficient 1, having minimal norm in the space
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...next to the knot $8_9$ is written $7-5+3-1$. This means that the Alexander polynomial equals $\Delta(t)=-t^6+3t^5-5t^4+7t^3-5t^2+3t-1$. Non-alternating knots are
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...l and its generalizations (e.g. the [[Jones–Conway polynomial|Jones–Conway polynomial]]).
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...the identity matrix. The multiplicity of an eigen value as a root of this polynomial is called its algebraic multiplicity. For any linear transformation of a fi
...ield $k$ (or a characteristic root of $A$) is a root of its characteristic polynomial.
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''separable game, polynomial-like game''
the degenerate game is called a polynomial game. In any two-person zero-sum degenerate game on the unit square player
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...is unipotent if and only if its [[Characteristic polynomial|characteristic polynomial]] is $ ( x - 1) ^ {n} $.
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...en over all black regions. The second Listing polynomial, or white Listing polynomial, $P _ { W } ( \delta , \lambda )$ is defined in a similar manner, summing o
...corners (e.g. to define the [[Kauffman bracket polynomial|Kauffman bracket polynomial]]), studying labelling of corners of alternating diagrams (e.g. to proof th
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...of a polynomial by a linear binomial: The remainder of the division of the polynomial
...e of Bezout's theorem is the following: A number $\alpha$ is a root of the polynomial $f(x)$ if and only if $f(x)$ is divisible by the binomial $x-\alpha$ withou
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A field $K$ for which every homogeneous polynomial form over $K$ of degree $d$ in $n$ variables with $n > d$ has a non-trivial
...strongly quasi-algebraically closed'' if the same properties holds for all polynomial forms. More generally, a field is $C_i$ if every form with $n > d^i$ has a
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...pt of a limit transition), and makes sense for any coefficient ring. For a polynomial
of a polynomial, then $ x _ {0} $
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...n )$-matrices $A$ such that the [[Characteristic polynomial|characteristic polynomial]] of $A$, $\operatorname { det } ( \lambda I - A )$, is equal to $f$. Indee
...and their minimal polynomial (cf. [[Minimal polynomial of a matrix|Minimal polynomial of a matrix]]) is $f$, i.e. their similarity invariants are $1 , \dots , f$
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A homogeneous polynomial of the first degree (cf. [[Homogeneous function]]).
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...ements for the variables. The ''standard polynomial'' of degree $n$ is the polynomial
...commutative ring satisfies the standard polynomial of degree $2n$, and no polynomial of lower degree.
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A polynomial $f\in k[x]$ is called separable over $k$ if none of its
a root of a polynomial that is separable over $k$. Otherwise $\a$ is
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...atrix is then a polynomial in the variables $x_{ij}$ and is non-zero (as a polynomial) if and only if a perfect matching exists.
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...r of the coefficients of a polynomial $\phi(x)$. Then $f(x)=d\phi(x)$ is a polynomial with integer coefficients; moreover, any factorization of $\phi(x)$ into ir
...ply, by the equations for the coefficients. It is then checked whether the polynomial $g(x)$ found in this way divides $f(x)$. This construction and the subseque
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A method for simultaneously calculating all roots of a polynomial. Suppose that the roots $ r _ {1} \dots r _ {n} $
of the polynomial
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It is a Laurent polynomial of one variable associated to ambient isotopy classes of unoriented framed
...^ { ( 1 ) } \rangle = - A ^ { 3 } \langle L \rangle$. The Kauffman bracket polynomial is also considered as an invariant of regular isotopy (Reidemeister moves:
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...$ is the root of a polynomial of the form $X^p - X - a$, an Artin–Schreier polynomial.
...ing Artin-Schreier polynomial has no root in $F$: it is an [[irreducible polynomial]] and the [[quotient ring]] $F[X]/\langle A_\alpha(X) \rangle$ is a field
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...polynomial of best approximation in the system $\{x_k\}$, that is, of the polynomial
...ernation) hold. The Haar condition is sufficient for the uniqueness of the polynomial of best approximation in the system $\{x_k\}_{k=1}^n$ with respect to the m
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...ave the same [[rank]], the same [[determinant]], the same [[characteristic polynomial]], and the same [[eigenvalue]]s. It is often important to select a matrix s
...expressed by the matrix $S$. The rank, determinant, trace, characteristic polynomial and so forth are properties of the endomorphism.
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A natural number [[Divisor (of an integer or of a polynomial)|divisor]] of $n$ other than $n$ itself.
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...ers vanish. If the last term on the right-hand side of (3) is omitted, the polynomial $ B _ {2n + 1 } (x _ {0} + th) $,
which is not a proper interpolation polynomial (it coincides with $ f(x) $
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...al]] for $n \leq 4$ and the [[Kauffman bracket polynomial|Kauffman bracket polynomial]] for $n \leq 3$. Also, the problem for which $n$ and $p$ a link and its $n
...eferences|[a1]]]. It is an open problem (as of 2000) whether the Alexander polynomial is preserved under rotation for any $n$, [[#References|[a3]]]. P. Traczyk h
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over a polynomial ring $ k [ x] $''
...characteristic polynomial, and their least common multiple is its minimum polynomial. Any collection of polynomials of the form $ l _ {i} ( x) = g _ {i} ( x)
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It is a Laurent polynomial of two variables associated to ambient isotopy classes of links in $\mathbf
...nding on whether the move is positive or negative). To define the Kauffman polynomial from $\Lambda _ { L } ( a , x )$ one considers an oriented link diagram $L
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A polynomial $f$ with coefficients in a field or a commutative associative ring $K$ with
...portant examples of symmetric polynomials are the ''[[elementary symmetric polynomial]]s''
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...ute a basis for the $A$-module $B$; finally, let $f(x)$ be the irreducible polynomial of $\theta$ over $k$, let $f^*(x)$ be the image of $f(x)$ in the ring $A/\m
with the degree of the polynomial $f_i^*(x)$ equal to the degree $[B/\mathfrak{P}_i : A/\mathfrak{p}]$ of the
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A representation of a function as a sum of its [[Taylor polynomial|Taylor polynomial]] of degree $n$ ($n=0,1,2,\dotsc$) and a remainder term. If a real-valued f
is its [[Taylor polynomial]], while the remainder term $r_n(x)$ can be written in Peano's form:
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...roblems in [[Algebraic geometry|algebraic geometry]] (commutative algebra, polynomial ideal theory) can be reduced by structurally easy algorithms to the constru
...lynomials with coefficients in certain rings, non-commutative polynomials, polynomial modules, and differential algebras. The algorithm that constructs Gröbner
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...c polynomial|trigonometric polynomial]] at some point by the values of the polynomial itself at a finite number of points. If $ T _ {n} ( x) $
is a trigonometric polynomial of degree $ n $
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...$x,y,\dots,w$ are variables and $A,B,\dots,D$ (the ''coefficients'' of the polynomial) and $k,l,\dots,t$ (the ''exponents of the powers'', which are non-negative
...with zero coefficients and, in each individual term, zero powers. When the polynomial has one, two or three terms it is called a monomial, binomial or trinomial.
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$#C+1 = 13 : ~/encyclopedia/old_files/data/L060/L.0600810 Lommel polynomial
The polynomial $ R _ {m, \nu } ( z) $
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...s an eigen value or characteristic value of a matrix (see [[Characteristic polynomial]]).
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...teger programming problems in a fixed number of variables can be solved in polynomial time.
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[[Characteristic polynomial|characteristic polynomial]] for the
dynamical system. The polynomial $w(z)$ (or, equivalently, the
4 KB (607 words) - 02:33, 14 September 2022
The points $x_j$ are the roots of the polynomial $P_n^{(1,1)}(x)$ (a Jacobi polynomial), orthogonal on $[-1,1]$ with respect to the weight $1-x^2$, $A=2/(n+1)(n+2
2 KB (237 words) - 20:21, 1 January 2019
...ords of weight one correspond to the generators $a_1,a_2,\ldots$. The Hall polynomial associated with the Hall element $t \in H$ is then computed in the [[free a
...ve this result combinatorially by first showing that any non-commutative [[polynomial]] is a sum of non-increasing products $P_{t_1}\cdots P_{t_n}$ (with non-neg
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...extension $L/K$; for this it is necessary and sufficient that the minimal polynomial has no multiple roots in the algebraic closure $\bar K$ of $K$, that is, th
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Suppose one is given a polynomial
...g four conditions is necessary and sufficient in order that all roots of a polynomial \eqref{*} with real coefficients have negative real parts:
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$#C+1 = 50 : ~/encyclopedia/old_files/data/P073/P.0703730 Polynomial of best approximation
A polynomial furnishing the best approximation of a function $ x ( t) $
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''polynomial ring''
...ials in an infinite set of variables if it is assumed that each individual polynomial depends only on a finite number of variables. A ring of polynomials over a
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A transformation of an $n$th degree polynomial equation
...that time (around 1683), these transformations do not help solving general polynomial equations of degree larger than four (see also [[Galois theory|Galois theor
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...bvarieties of a projective space with a given [[Hilbert polynomial|Hilbert polynomial]] can be endowed with the structure of an [[Algebraic variety|algebraic var
...e fibre $Z_{s^*}$ of the projection of $Z$ on $S^*$ has $P$ as its Hilbert polynomial. The functor $\operatorname{Hilb}_{S/X}^P$ can be represented by the Hilber
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...g a Routh scheme, the number of complex roots with positive real part of a polynomial $ f ( x) $
The Routh scheme of this polynomial is defined to be the array of numbers
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...$ which is monic (with leading coefficient equal to 1), irreducible in the polynomial ring $k[x]$ and satisfying $f_\a(\a) = 0$; any
polynomial over $k$ having $\a$ as a root is divisible by $f_\a(x)$. This
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for every [[Trigonometric polynomial|trigonometric polynomial]] $ Q $
for every (algebraic) [[Polynomial|polynomial]] $ Q $
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...ta$, the resulting Chebyshev function is truly an $n$th order [[Polynomial|polynomial]] in $x$, but it is also a cosine function with a change of variable. Thus,
...formula]]), if the interpolation points are taken to be the zeros of this polynomial, the error is minimized. A related and possibly more useful set of interpol
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...family of polynomials with coefficients in $K$ (cf. [[Splitting field of a polynomial]]);
3) any polynomial $f(x)$ with coefficients in $K$, irreducible over $K$ and having a root in
2 KB (288 words) - 18:30, 11 April 2016
...een a continuous function $f(x)$ on a closed set of real numbers $Q$ and a polynomial $P_n(x)$ (in a [[Chebyshev system|Chebyshev system]] $\{\phi_k(x)\}_0^n$) o
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...Jones–Conway polynomials (cf. also [[Jones–Conway polynomial|Jones–Conway polynomial]]) and the same Murasugi signatures (for links with non-zero determinant, c
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is a real closed field) is a set that can be given by finitely many polynomial equalities and inequalities. More precisely, for $ g \in \mathbf R [ X _
...deciding the truth of any elementary sentence built up from finitely many polynomial inequalities $ g _ {i} ( x _ {1}, \dots, x _ {n} ) > 0 $,
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where $Q$ is the splitting field of the polynomial $fg$ (cf.
[[Splitting field of a polynomial|Splitting field of a polynomial]]), and $\a_i,\b_j$ are the roots (cf.
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...Bézier representation overcomes numerical and geometric drawbacks of other polynomial forms. Bézier curves and surfaces were independently developed by P. de Ca
Every polynomial of degree $ \leq n $
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...ed using the constant term $\phi(I,0)$ of the [[Hilbert polynomial|Hilbert polynomial]] $\phi(I,m)$ of $I$ by the formula
...the virtual arithmetic genus $p_a(D)$ as the constant term of the Hilbert polynomial of the coherent sheaf $\mathcal O_X(D)$ corresponding to $D$. If the diviso
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A [[first integral]] of a polynomial vector field on the plane, which has a specific form, the product of (non-i
...only isolated singularities (i.e., $\gcd(P,Q)=1$); denote by $\omega$ the polynomial $1$-form $-Q(x,y)\rd x+P(x,y)\rd y$ annulating $v$, so that $\omega\cdot v\
5 KB (809 words) - 09:08, 12 December 2013
Let $f(z)$ be a complex polynomial, i.e., $f(z)\in\mathbf C[z]$. Then the zeros of the derivative $f'(z)$ are
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such that for any polynomial
If for any such polynomial $ \Phi ( P) > 0 $,
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...in one variable $x$ and $f(x)$ is an [[Irreducible polynomial|irreducible polynomial]]. This quotient ring describes all field extensions of $F$ by roots of the
2 KB (270 words) - 14:56, 30 December 2018
There exists a polynomial $ t _ {n} ( z) \in K _ {n} $
it is called the Chebyshev polynomial for $ E $.
4 KB (516 words) - 06:48, 22 February 2022
...inite-dimensional or, equivalently, if the element $a$ has an annihilating polynomial with coefficients from the ground field $F$. An algebra $A$ is called an ''
...fields are commutative. An algebraic algebra of bounded degree satisfies a polynomial identity (cf. [[PI-algebra]]). An algebraic PI-algebra is locally finite. I
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...eral strategy for computing linear and cyclic convolutions by applying the polynomial version of the [[Chinese remainder theorem|Chinese remainder theorem]] [[#R
For any polynomial $ m ( x ) $
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For each $\sigma \in \mathbf{R}$, the Zolotarev polynomial $Z _ { n } ( x ; \sigma )$ is the unique solution of the problem
...roximation|Uniform approximation]]; [[Polynomial least deviating from zero|Polynomial least deviating from zero]]).
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The problem of extending a polynomial in $ z $
in the class of all regular functions in the unit disc having the given polynomial as initial segment of the MacLaurin series. The solution to this problem is
3 KB (480 words) - 06:30, 30 May 2020
A complex (sometimes, real) number that is a root of a polynomial
...ger, is an algebraic number of degree $n$, being a root of the irreducible polynomial $x^n-2$.
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It is desirable that such a test run in polynomial time; that is, the number of bit operations used to test $ n $ for primalit
...computer science because it is not known whether there is a deterministic polynomial-time algorithm for primality (cf. [[Complexity theory]]). In addition, they
3 KB (485 words) - 22:01, 25 October 2014
...{ i } $ of $Q _ { 2 ^{ i} ( n + 1 ) - 1 }$ are precisely the zeros of the polynomial $E _ { 2 ^{i-1}(n+1)} ^ { i } $ which satisfies
...= \sqrt { 1 - x ^ { 2 } } / \rho _ { m } ( x )$, where $\rho _ { m }$ is a polynomial of degree $m$ which is positive on $[ a , b ] = [ - 1,1 ]$, see [[#Referenc
5 KB (655 words) - 06:08, 15 February 2024
is a polynomial in $ n \geq 2 $
is an absolutely-irreducible polynomial with integer rational coefficients, then for the number $ N _ {p} $
4 KB (568 words) - 17:46, 4 June 2020
...lynomial and exponential growth in groups and algebras]]), and if it is of polynomial growth, then it is polycyclic and almost nilpotent (i.e. it contains a subg
4 KB (588 words) - 19:59, 11 April 2014
...obian conjecture. This conjecture is still open (1999) for all $n \geq 2$. Polynomial mappings satisfying $\operatorname{det} JF \in \mathbf{C}^*$ are called Kel
a) up to a polynomial coordinate change, $( \partial _ { 1 } , \dots , \partial _ { n } )$ is the
7 KB (1,112 words) - 05:20, 15 February 2024
...r or not a given integer is prime; and ii) the problem of deciding for any polynomial with integer coefficients whether or not it has a real root.
...ny multi-variable polynomial with integer coefficients whether or not that polynomial has all integer roots ( "Hilbert 10th problem", cf. [[Hilbert problems]]);
3 KB (490 words) - 22:33, 10 January 2016
With the convention $c_0=-1$, one defines the feedback polynomial of the LFSR as
its reciprocal polynomial
9 KB (1,477 words) - 09:28, 17 May 2021
One then chooses a value of $\a$ such that the quadratic [[Polynomial|trinomial]] in
roots of this equation. For $\a=\a_0$ the polynomial in square brackets in
2 KB (271 words) - 13:05, 17 December 2014
can be found without knowing the decomposition (1) of the polynomial $ Q( x) $
into irreducible factors: The polynomial $ Q _ {1} ( x) $
3 KB (482 words) - 15:56, 2 March 2022
The nodes are the roots of the Chebyshev polynomial
and hence the other nodes of which are the roots of the orthogonal polynomial of degree $ 2N- 1 $
3 KB (490 words) - 17:39, 14 December 2020
...ta$ is a Salem number, then it is reciprocal in the sense that its minimal polynomial $P(x)$ satisfies $P(x) = x^d P(1/x)$, where $d$ is the degree of $P$, so $d
...known as Lehmer's number. The minimum polynomial of $\sigma_1$ is Lehmer's polynomial: $x^{10} + x^9 - x^7 - x^6 - x^5 - x^4 -x^3 + x + 1$. This is also the smal
3 KB (504 words) - 18:43, 14 April 2023
is a polynomial $ \mathop{\rm mod} 2 $,
a so-called Zhegalkin polynomial, named after I.I. Zhegalkin, who initiated the investigation of this clone
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A parametric polynomial defined by a degree $n$, and a sequence of $n + 1$ control points $P_{0}, \
...representation and do not have the flexibility and generality of piecewise-polynomial representations such as $ B $-
5 KB (759 words) - 09:49, 27 March 2023
...uotient algebra $F[X]/T$ is a [[PI-algebra|PI-algebra]] with $T$ as set of polynomial identities. It is called the relatively free algebra (or generic algebra) w
...<TR><TD valign="top">[1]</TD> <TD valign="top"> C. Procesi, "Rings with polynomial identities" , M. Dekker (1973)</TD></TR><TR><TD valign="top">[2]</TD> <TD
4 KB (732 words) - 08:33, 29 August 2014
A formula for obtaining a polynomial of degree $ n $(
the Lagrange interpolation polynomial) that interpolates a given function $ f ( x) $
4 KB (640 words) - 14:37, 13 January 2024
...but a subring of a Noetherian ring need not be Noetherian. For example, a polynomial ring in infinitely many variables over a field is not Noetherian, although
is a left Noetherian ring, then so is the polynomial ring $A[X]$.
3 KB (539 words) - 18:51, 3 April 2024
times the leading coefficient of the Hilbert–Samuel polynomial of $ A $,
...logical discrepancies in the literature with respect to the Hilbert–Samuel polynomial. Let $ \psi ( n) = \textrm{length} _ {A} ( M / \mathfrak a ^ {n+ 1} M )
4 KB (548 words) - 06:47, 16 June 2022
A method for computing the value at a point $x$ of the interpolation polynomial $L_n(x)$ with respect to the nodes $x_0,\ldots,x_n$, based on the successiv
where $L_{(i,\ldots,m)}(x)$ is the interpolation polynomial with interpolation nodes $x_i,\ldots,x_m$, in particular, $L_{(i)}(x) = x_i
2 KB (335 words) - 14:03, 30 April 2023
...anslation" is also used: as are "algebraic function" (of one variable) or "polynomial".
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A [[Linear operator|linear operator]] satisfying a [[Polynomial|polynomial]] identity with scalar coefficients.
...]). Note that I. Kaplansky in [[#References|[a1]]] considered rings with a polynomial identity (cf. also [[PI-algebra|PI-algebra]]).
10 KB (1,502 words) - 22:40, 12 December 2020
be a polynomial in $ X _ {1} \dots X _ {n} $
of zeros of this polynomial can be regarded as the graph of a correspondence $ y : \mathbf C ^ {n}
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...am. It has its root in the statistical mechanics model of the Jones–Conway polynomial by V.F.R. Jones. It has been applied to periodic links and to the building
...ent to work with the following regular isotopy variant of the Jones–Conway polynomial:
5 KB (744 words) - 05:30, 15 February 2024
...inatorial analysis]]), and to Bell polynomials (cf. [[Bell polynomial|Bell polynomial]]) by
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...an identity. A root of this equation is also called a root or zero of the polynomial
...one root (hence as many roots as its degree, counting multiplicities). The polynomial $f(x)$ may be expressed as a product
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[[Hilbert polynomial|Hilbert polynomial]] $h(n)=\chi(V,\cL^n)$ such that for $n>c$ the
sheaves $\cL_S^n$ with Hilbert polynomial $h(n)$ and with $H^i(X_s,\cL^n_S)=0$ for $i>0$, are very
4 KB (644 words) - 13:06, 17 April 2023
Let $R(\cdot ,\cdot)$ be a rational function of two variables and $f(z)$ a polynomial of degree three or four, without multiple roots. A pseudo-elliptic integral
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frequently occur as interpolation nodes in quadrature formulas. The polynomial $ T _ {n} ( x) $
that is, for any other polynomial $ \widetilde{F} _ {n} ( x) $
5 KB (690 words) - 16:20, 6 January 2024
The simplest form of an algebraic expression, a [[Polynomial|polynomial]] containing only one term.
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This is a generalization of the classical concept of a [[polynomial of best approximation]]. The main questions concerning elements of best app
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...ystems, variety of|Algebraic systems, variety of]]). Since the totality of polynomial identities that are satisfied in a given ring forms a fully-characteristic
<TR><TD valign="top">[1]</TD> <TD valign="top"> C. Procesi, "Rings with polynomial identities" , M. Dekker (1973)</TD></TR>
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is called split or splitting if the characteristic polynomial of each of the transformations has all its roots in $ k $,
...d of a polynomial|Splitting field of a polynomial]]) of the characteristic polynomial of each $ h \in L $.
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of formula (1) are the roots of a polynomial of degree $ N $
if and only if it is an interpolatory quadrature formula and the polynomial $ \omega ( x) $
6 KB (985 words) - 08:05, 9 February 2024
is the degree of the Alexander polynomial (cf. [[Alexander invariants|Alexander invariants]]), $ h $
and the reduced Alexander polynomial of seperated links is zero. The Alexander matrix is computed as the inciden
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...f algebra]] and the references quoted there. See also [[Lie polynomial|Lie polynomial]] for the concrete case that $ L $
2 KB (345 words) - 17:37, 16 December 2020
...sometimes called a Hilbert algebra. The theorem that a finitely-generated polynomial-identity algebra (cf. [[PI-algebra]]) over a field is a Hilbert algebra, is
<TR><TD valign="top">[a1]</TD> <TD valign="top"> C. Procesu, "Rings with polynomial identities" , M. Dekker (1973)</TD></TR>
3 KB (415 words) - 20:32, 19 January 2016
For the content of a polynomial, see [[Primitive polynomial]].
2 KB (378 words) - 15:57, 8 October 2017
...t can be decided by an algorithm the running time of which is bounded by a polynomial function in the input size.
...ss $\overline{z}$ such that $M$ on input $( x , \overline{z} )$ accepts in polynomial time with respect to $\operatorname{size}( x )$. If $x \notin S$, no $z$ ex
7 KB (1,161 words) - 16:46, 1 July 2020
...such that for each element of the space under consideration there exists a polynomial in this operator (with scalar coefficients) annihilating this element.
...hat $p ( T ) x = 0$ (cf. [[#References|[a1]]]). If there exists a non-zero polynomial $p ( t ) \in \mathbf{F} [ t ]$ such that $p ( T ) x = 0$ for every $x \in X
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is (representable as) a piecewise-polynomial function on the linear hull $ { \mathop{\rm ran} } \Xi $
of its directions, with support in the convex hull of its directions, its polynomial degree being equal to $ s - { \mathop{\rm dim} } { \mathop{\rm ran} } \Xi
2 KB (339 words) - 06:29, 30 May 2020
...[Matroid|matroid]] with rank function $r$ on the ground set $E$. The Tutte polynomial $t ( M ; x , y )$ of $M$ is defined by
Some standard evaluations of the Tutte polynomial are:
11 KB (1,736 words) - 06:27, 15 February 2024
..., cf. [[#References|[3]]]). The isolated prime ideals of an ideal $I$ of a polynomial ring over a field correspond to the irreducible components of the [[affine
2 KB (386 words) - 20:06, 5 October 2017
...braic number field (of degree $n$) if every $\alpha\in K$ is the root of a polynomial (of degree at most $n$) over $\mathbf Q$. (Cf. also [[Algebraic number]];
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...of the uniform norm of the derivative in terms of the uniform norm of the polynomial itself. Let $ P _ {n} ( x) $
be an algebraic polynomial of degree not exceeding $ n $
3 KB (399 words) - 10:55, 6 February 2021
where $f(x)$ is a polynomial in $x$ and
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...lign="top">[4]</TD> <TD valign="top"> D. Quillen, "Projective modules over polynomial rings" ''Invent. Math.'' , '''36''' (1976) pp. 167–171 {{MR|0427303}} {{Z
is a monic polynomial such that $ M _ {f} $
7 KB (1,080 words) - 08:08, 6 June 2020
...rtition $\Delta_n$: $\alpha=x_0<x_1<\cdots<x_n=b$ with a certain algebraic polynomial of degree at most $m$. Splines can be represented in the following way:
where the $c_k$ are real numbers, $P_{m-1}(x)$ is a polynomial of degree at most $m-1$, and $(x-t)^m_{+}=\max\left(0,(x-t)^m\right)$.
3 KB (503 words) - 08:54, 3 May 2012
...or, equivalently, $\mathcal{NC}$ is the class of languages recognizable in polynomial size and poly-log-depth circuits [[#References|[a5]]]. In fact, many combin
...C}$, where $\mathcal P$ consists of all the problems that can be solved in polynomial sequential time (cf. also [[Complexity theory|Complexity theory]]; [[NP|$\m
6 KB (940 words) - 23:19, 25 November 2018
The second Euler substitution: If the roots $x_1$ and $x_2$ of the quadratic polynomial $ax^2+bx+c$ are real, then
...mptotes of this hyperbola; when the roots $x_1$ and $x_2$ of the quadratic polynomial $ax^2+bx+c$ are real, the second Euler substitution is obtained by taking a
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A [[Trigonometric polynomial|trigonometric polynomial]] is an expression in one of the equivalent forms $ a _ {0} + \sum _ {1}
When the values of a trigonometric polynomial are real for all real $ t $,
9 KB (1,334 words) - 13:15, 26 March 2023
is a homogeneous polynomial of degree $ m $,
while the polynomial $ F $
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...thrm{GF}(p)(\alpha)$. Such an $\alpha$ will be any root of any irreducible polynomial of degree $n$ from the ring $\mathrm{GF}(p)[X]$. The number of primitive el
...order $p^n-1$, i.e. each element of $\mathrm{GF}(p^n)^*$ is a root of the polynomial $X^{p^n-1}-1$. The group $\mathrm{GF}(p^n)^*$ is cyclic, and its generators
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...commutative ring. A necessary condition is that it satisfies all universal
polynomial identities $p[x_1,\ldots,x_m]=0$ of the $n\times n$ matrix ring over the in
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> L.H. Rowen, "
Polynomial identities in ring theory" , Acad. Press (1980) pp. Chapt. 7</TD></TR></t
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...mpact case this sum is finite, since the critical points are discrete. The polynomial $ M _ {t} ( f ) $,
which is also called the Morse polynomial of $ f $,
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[[Kharitonov polynomial theory]] |
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be a [[separable polynomial]] over a field $ k $
which belongs to the splitting field of the polynomial $ f( x) $(
7 KB (1,042 words) - 19:53, 16 January 2024
...l time, that is, in a number of computational operations that depends as a polynomial on the so-called "input size" of the problem. The class $\mathcal{NP}$ incl
...e general [[Linear programming|linear programming]] problem is solvable in polynomial time.
5 KB (700 words) - 19:16, 4 November 2014
The class of functions consisting of the [[Polynomial|polynomials]], the [[Exponential function, real|exponential functions]], th
1 KB (146 words) - 15:13, 4 May 2012
...les that satisfies the [[Laplace equation|Laplace equation]]. Any harmonic polynomial may be represented as the sum of homogeneous harmonic polynomials. If $n=2$
2 KB (365 words) - 13:05, 14 February 2020
...$n \in \mathbf{N} _ { 0 } = \{ 0,1,2 , \dots \}$, the problem of finding a polynomial $p ( x , y )$ of degree $n$ which minimizes the $L^{2}$-norm
...N} _ { 0 }$ and $m = n - 2 j$ with $j = 0 , \dots , n$, the Zernike circle polynomial is
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...] (at $t=e^{2\pi i/6}$), and the group of $5$-colourings by the [[Kauffman polynomial]] (at $a=1$, $z = 2\cos(2\pi/5)$), [[#References|[a2]]]. The $n$-moves pres
2 KB (367 words) - 18:07, 14 November 2023
periodic also. For polynomial splines of degree $ 2k+ 1 $,
Spline interpolation has some advantages when compared to polynomial [[Interpolation|interpolation]]. E.g., there are sequences of partitions $
5 KB (751 words) - 08:22, 6 June 2020
if for any unitary polynomial $ P( X) \in A[ X] $
of the reduced polynomial $ \overline{P}\; ( X) $
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is a [[Polynomial|polynomial]] of degree $ n $
...entity, are known as the roots of the equation (1), or as the roots of the polynomial
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''quasi-symmetric polynomial (in combinatorics)''
...tative ring]] $R$ with unit element in the commuting variables from $X$. A polynomial or power series $f(X) \in R[[X]]$ is called ''symmetric'' if for any two fi
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Given a [[Polynomial|polynomial]] $P ( x _ { 1 } , \ldots , x _ { n } )$ with complex coefficients, the log
The Mahler measure is useful in the study of polynomial inequalities because of the multiplicative property $M ( P Q ) = M ( P ) M
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...e $n$ traces $\tr A, \ldots \tr A^n$ uniquely determine the characteristic polynomial of $A$. In particular, $A$ is nilpotent if and only if $\tr A^m = 0$ for al
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A generalization of a polynomial algebra. If $M$ is a [[unital module]] over a commutative associative ring
...\mapsto X_i$ ($i=1,\ldots,n$) extends to an isomorphism of $S(M)$ onto the polynomial algebra $A[X_1,\ldots,X_n]$ (see [[Ring of polynomials]]).
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...^ { n }$. (For $n > 1$ and $m > 1$ there is no unique interpolating polynomial of degree $\leq m$.)
...ifferentiable function on the convex hull of $p$, the Kergin interpolating polynomial $K _ { p } ( f )$ is of degree $\leq m$ and satisfies:
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...thm $f$ is subjected to the further requirement that it can be computed in polynomial time or even logarithmic space.
...align="top"> R.E. Ladner, N.A. Lynch, A.L. Selman, "A comparison of polynomial time reducibilities" ''Theor. Comp. Sc.'' , '''1''' (1975) pp. 103–123
3 KB (414 words) - 21:57, 16 January 2016
Since every polynomial $ a( x) $
to one and only one polynomial of the form
4 KB (700 words) - 18:53, 18 January 2024
...s n)$-matrix $F$, called a feedback matrix, such that the [[characteristic polynomial]] of $A+BF$ is precisely $(X-r_1)\cdots(X - r_n)$? The pair $(A,B)$ is then
...$ there is an $(m\times n)$-matrix $F$ such that $A+BF$ has characteristic polynomial $X^n + a_1X^{n-1} + \cdots + a_n$.
5 KB (799 words) - 20:38, 12 November 2017
...onsidered unlikely that an exact solution can be found for this problem in polynomial time and approximate solutions are looked for instead.
...]]], whereas in the Euclidean case the optimal tour can be approximated in polynomial time to within a factor of $1.5$ [[#References|[a4]]], p. 162, and, if $r =
4 KB (704 words) - 12:30, 17 February 2021
...deg } \omega ( z ) < \operatorname { deg } \sigma ( z ) \leq t$, given the polynomial $S ( z )$, $\operatorname { deg } S ( z ) < 2 t$. Originally intended for t
...a ( z )$, the error evaluator polynomial. The zeros of the error evaluator polynomial yield the coordinate positions at which errors occur and the error value at
8 KB (1,155 words) - 18:48, 26 January 2024
is a Bernoulli polynomial (cf. [[Bernoulli polynomials]]). The periodic functions coinciding with the
Using the same symbolic notation one has for every polynomial $ p( x) $,
3 KB (477 words) - 08:36, 6 January 2024
...sentation $|x,y\colon yx^{-1}yxy^{-1}=x^{-1}yxy^{-1}x|$, and the Alexander polynomial is $\Delta_1=t^2-3t+1$. It was considered by I.B. Listing [[#References|[1]
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is a simple positive root of the [[Characteristic polynomial|characteristic polynomial]] of $ A $;
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''Homfly polynomial, Homflypt polynomial, skein polynomial''
...[[Alexander–Conway polynomial|Alexander–Conway polynomial]] and the Jones polynomial.
18 KB (2,713 words) - 05:14, 15 February 2024
...cteristic zero is nilpotent (Higman's theorem); a nil algebra satisfying a polynomial identity is locally nilpotent. It is not clear (1982) whether a finitely-ge
...y, and $k$ is infinite. The radical of a finitely-generated algebra with a polynomial identity over a field of characteristic zero is nilpotent. This is equivale
6 KB (1,020 words) - 16:48, 25 March 2023
of $x_1^{n_1}\dotsm x_m^{n_m}$ in the expansion of the polynomial $(x_1+\dotsb+x_m)^n$. In combinatorics, the multinomial coefficient express
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...s possible to find a corresponding equation (*) in which the degree of the polynomial $ P $
...nomial with integer coefficients, for arbitrary values of the variables. A polynomial of degree at most 6 may be taken as $ P $
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The difference between the function $x^n$ and a polynomial [[Spline|spline]] $S_{n-1}(x)$ of degree $n-1$. Monosplines arise in the st
1 KB (173 words) - 10:16, 20 September 2014
...Lasker [[#References|[1]]] proved that there is a primary decomposition in polynomial rings. E. Noether [[#References|[2]]] established that any [[Noetherian rin
1 KB (157 words) - 11:43, 29 June 2014
...[[#References|[a3]]] proved a variant of an interior-point method to have polynomial worst-case complexity when applied to the linear programming problem. There
The analysis that shows which properties make it possible to prove polynomial convergence of interior-point methods for classes of convex programming pro
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...are related to symmetric polynomials (cf. [[Symmetric polynomial|Symmetric polynomial]]). Every rational symmetric function (over a field of characteristic 0) is
The theorem that a symmetric polynomial is a polynomial in the elementary symmetric functions is also known as Newton's theorem. Si
5 KB (731 words) - 07:17, 24 March 2023
...d that $\nu\leq(n/2)+1$. Thue's method is based on properties of a special polynomial $f(x,y)$ of two variables $x,y$ with integer coefficients, and the hypothes
...hod to the case of a polynomial in any number of variables, similar to the polynomial $f(x,y)$, and making use of the large number of solutions of \eqref{1}. The
4 KB (634 words) - 15:17, 14 February 2020
...x $z$-plane, known as the Nyquist diagram. Suppose that the characteristic polynomial $N(z)$ of the open-loop system has $k$, $0\leq k\leq n$, roots with positiv
4 KB (619 words) - 13:06, 10 August 2014
...e numerous other ways in which the word degree is used. E.g. degree of a [[polynomial]]; [[degree of a mapping]]; degree of unsolvability, degree of irrationalit
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...ots,n-1$, then the roots of the [[Characteristic polynomial|characteristic polynomial]] of $J$ are real and distinct.
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...r obtaining the incomplete fraction and the remainder in the division of a polynomial
...ots,a_n$ lie in a certain field, e.g. in the field of complex numbers. Any polynomial $f(x)$ can be uniquely represented in the form
3 KB (445 words) - 18:09, 1 October 2017
is a polynomial, then $ S $
is called a [[Weyl sum|Weyl sum]]; if the polynomial $ F $
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...from Weyl's criterion and his estimates for trigonometric sums involving a polynomial $f$,
...,1)$ provided that at least one coefficient $a_s$, $1\leq s\leq k$, of the polynomial
3 KB (454 words) - 19:13, 13 November 2023
and may be extended to a polynomial function on $ \mathop{\rm End}\nolimits \ V $.
is sometimes applied to a polynomial function on $ \mathop{\rm End}\nolimits \ V $
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non-zero polynomial with coefficients in $k$.
$k$ if for any finite set $x_1,\dots,x_m \in X$ and any non-zero polynomial $F(X_1,\dots,X_m)$ with
3 KB (492 words) - 19:41, 13 December 2015
...|[a3]]], [[#References|[a4]]]) solves the following problem concerning the polynomial ring $ R [ {\mathcal X} ] $
2) Given a finite set of polynomial equations over $ R $
10 KB (1,471 words) - 14:52, 30 May 2020
is zero, then all irreducible polynomial linear representations of these groups can be realized by means of tensors.
all (differentiable) linear representations are polynomial; every linear representation of $ \textrm{ GL} ( V) $
14 KB (1,916 words) - 08:00, 5 May 2022
be a given [[Polynomial|polynomial]]. The Carathéodory–Schur extension problem is to find (if possible) an
one sees that the Carathéodory–Schur extension problem for the polynomial $ a _ {0} + a _ {1} z + \dots + a _ {n - 1 } z ^ {n - 1 } $
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...solution $u$ is approximated by a [[Trigonometric
polynomial|trigonometric
polynomial]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org
The Lagrange interpolation
polynomial (cf. also [[Lagrange interpolation formula|Lagrange interpolation formula]]
7 KB (1,028 words) - 17:44, 1 July 2020
A method of writing the interpolation polynomial obtained from the [[Gauss interpolation formula|Gauss interpolation formula
Compared with other versions of the interpolation polynomial, formula (1) reduces approximately by half the amount of work required to s
4 KB (592 words) - 08:09, 26 November 2023
For a fixed $n$ there is a polynomial-time algorithm to solve the Frobenius problem, [[#References|[a1]]].
1 KB (154 words) - 21:17, 8 April 2018
...ations of the indices of variables $x_1,\ldots,x_n$, leave the alternating polynomial $\prod(x_i-x_j)$ invariant, hence the term "alternating group". The group
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splitting field of a polynomial $f$ over $k$, the Galois group $G(L/k)$ is
also called the Galois group of the polynomial $f$. These groups are
3 KB (494 words) - 21:56, 5 March 2012
In order for the interpolation polynomial (2) to exist for any function $ f ( x) $
The interpolation polynomial will, moreover, be unique and its coefficients $ a _ {i} $
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...; \lambda )$ of a [[ranked poset|ranked partially ordered set]] $L$ is the polynomial
...nction $\phi ( G ; s )$ of a finite group $G$, defined to be the Dirichlet polynomial
11 KB (1,758 words) - 09:24, 10 November 2023
...non-varying elements in all kinds of expressions, e.g. the constants of a polynomial (also called its coefficients, cf. [[Coefficient]]), field constants (when
1 KB (153 words) - 21:11, 20 December 2015
say; the polynomial $ r $
the polynomial coefficients $ A _ {i} $
6 KB (908 words) - 06:04, 12 July 2022
...r $A$ (cf. [[Lie algebra, free|Lie algebra, free]]; [[Hall polynomial|Hall polynomial]]). X. Viennot [[#References|[a1]]] has shown that his definition not only
4 KB (628 words) - 18:40, 5 October 2023
...algebraic over $K$ is a purely inseparable element: that is, has a minimal polynomial of the form $X^{p^e} - a$ where $a \in K$.
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...e all in the open unit disc $\mathbf D$ (cf. also [[Szegö polynomial|Szegö polynomial]]). Therefore, the para-orthogonal polynomials are introduced as $Q _ { n }
3 KB (454 words) - 16:59, 1 July 2020
...tion about the [[autocorrelation]] of binary sequences. See [[Littlewood polynomial]].
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...9]]], [[#References|[a10]]], [[#References|[a11]]] for applications of the polynomial Hales–Jewett theorem to density Ramsey theory.
...rences|[a14]]] and [[#References|[a15]]]. For an infinitary version of the polynomial Hales–Jewett theorem, see [[#References|[a7]]], Sect. 2.6.
10 KB (1,506 words) - 00:47, 15 February 2024
...ession is the solution of a linear difference equation with characteristic polynomial $ (x - x _ {p} ) ^ {p} $,
as characteristic polynomial, and (a1) must be replaced by the corresponding difference equation of orde
3 KB (496 words) - 09:54, 29 May 2020
A restriction of a homogeneous [[Harmonic polynomial|harmonic polynomial]] $h^{(k)}(x)$ of degree $k$ in $n$ variables $x=(x_1,\dots,x_n)$ to the un
3 KB (519 words) - 19:48, 22 December 2018
...rivative in terms of the polynomial itself. If $T_n(x)$ is a trigonometric polynomial of degree not exceeding $n$ and if
...ty for an algebraic polynomial has the following form {{Cite|Be2}}: If the polynomial
5 KB (873 words) - 17:30, 24 July 2012
...quiv1$. He obtained the following formula, which is exact for an arbitrary polynomial of degree not exceeding $2n-1$:
where the $x_k$ are the roots of the Legendre polynomial (cf. [[Legendre polynomials|Legendre polynomials]]) $P_n(x)$, while $A_k^{(
4 KB (614 words) - 21:36, 1 January 2019
...} ( u ^ { \lambda } ) = \pi ( \lambda ) z ^ { \lambda }$ with the indicial polynomial
In the following, the zeros $\lambda _ { i }$ of the indicial polynomial will be ordered by requiring
13 KB (1,803 words) - 07:40, 4 February 2024
...lgebraic over $k$; that is, every element of $K$ is the root of a non-zero polynomial with coefficients in $k$. A finite degree extension is necessarily algebra
1 KB (190 words) - 14:18, 12 November 2023
The sequence of values of a polynomial of degree $m$:
assumed by the polynomial when the variable $x$ takes successive integral non-negative values $x=0,1,
4 KB (566 words) - 07:28, 26 March 2023
...op">[a1]</TD> <TD valign="top"> E.J. Ditters, A.C.J. Scholtens, "Free polynomial generators for the Hopf algebra $\mathit{Qsym}$ of quasi-symmetric function
1 KB (182 words) - 19:16, 17 June 2016
is the Laguerre polynomial (cf. [[Laguerre polynomials]]) of degree $ n $.
is the generalized Laguerre polynomial (see [[#References|[4]]]).
3 KB (477 words) - 08:39, 6 January 2024
where $M$ is a finite set of natural numbers, is called the Dirichlet polynomial with coefficients $a _ { m }$ (complex numbers) and exponents $\lambda _ {
...esis (cf. [[Riemann hypotheses|Riemann hypotheses]]) is that the Dirichlet polynomial $\sum _ { { m } = 1 } ^ { { n } } m ^ { - s }$ should have no zeros in $\
5 KB (706 words) - 17:03, 1 July 2020
is a polynomial in the variable $ t $,
called the chromatic polynomial of $ G $.
5 KB (799 words) - 20:12, 15 March 2023
...s rational (cf. [[Polynomial and exponential growth in groups and algebras|Polynomial and exponential growth in groups and algebras]]). Every hyperbolic group is
3 KB (519 words) - 09:13, 6 September 2014
The polynomial $ L _ {n} ^ \alpha ( x) $
The Laguerre polynomial $ L _ {n} ^ \alpha ( x) $
4 KB (571 words) - 18:26, 31 August 2020
All roots of a polynomial
with a constant matrix $A$, the characteristic polynomial of which is $P ( z )$ (see [[#References|[4]]]).
5 KB (692 words) - 15:30, 1 July 2020
...is $P ( z , f ( z ) , f ( z ^ { d } ) ) = 0$, where $P$ is a [[Polynomial|polynomial]] with algebraic coefficients and $d \geq 2$ an integer. For instance, the
...y polynomial. This construction is different from Hermite's one, since the polynomial is not explicit, and also different from Siegel's, Gel'fond's or Schneider'
6 KB (816 words) - 20:26, 5 December 2023
$#C+1 = 103 : ~/encyclopedia/old_files/data/B110/B.1100250 Bell polynomial
is a homogeneous polynomial of degree $ k $
12 KB (1,714 words) - 10:58, 29 May 2020
one obtains a way of writing the polynomial $ L _ {n} ( x) $
...nterpolation. If the same change of variables is made in the interpolation polynomial $ L _ {n} $
4 KB (623 words) - 07:44, 14 January 2024
is a polynomial of degree $(m-1)$ with respect to $n$.
1 KB (185 words) - 16:46, 23 November 2023
...A. Lascoux and M.-P. Schützenberger [[#References|[a18]]] as distinguished polynomial representatives of Schubert cycles (cf. also [[Schubert cycle|Schubert cycl
...est element), successive application of divided difference operators gives polynomial representatives of all Schubert cycles, which are independent of the choice
10 KB (1,494 words) - 09:23, 10 November 2023
The problem of polynomial approximation is a particular case of the problem of approximation by ratio
Mergelyan's theorem touches upon a large number of papers concerning polynomial, rational and holomorphic approximation in the space $\mathbf C^n$ of sever
5 KB (707 words) - 19:02, 1 May 2014
Let $P$ be a [[Polynomial|polynomial]] in $n$ variables, $P ( \xi ) = \sum _ { J } a _ { J } \xi ^ { J }$, where
...solutions, functions of the type $Q ( x ) e ^ { i \xi x }$, where $Q$ is a polynomial and necessarily $P ( \xi ) = 0$. See [[#References|[a2]]], [[#References|[a
8 KB (1,264 words) - 05:21, 19 March 2022
...polynomial]] of each symplectic automorphism of $E$ is an Euler–Frobenius polynomial $p _ { m } ( x )$ of odd degree $m - 1$.
5 KB (742 words) - 07:27, 25 January 2024
...ral circulant with first row $(a_0, a_1, \ldots, a_{n-1})$ is equal to the polynomial $a(C) = a_0 I + a_1 C + \cdots + a_{n-1} C^{n-1}$. Hence all circulant mat
1 KB (160 words) - 19:40, 17 November 2023
...show that the only possible limit sets for vector fields on the sphere<ref>Polynomial vector fields on the plane also satisfy this assertion.</ref> are singular
A polynomial planar vector field after [[complexification]] defines a holomorphic singul
7 KB (1,059 words) - 09:14, 1 October 2023
The $m$-th Todd polynomial $T_m(c_1,\ldots,c_m)$ is defined by $T_m(\sigma_1,\ldots,\sigma_m)$ being t
1 KB (169 words) - 13:51, 4 November 2023
...s (or non-negative integers or positive integers) for which there exists a polynomial $P(a_1,\ldots,a_n,z_1,\ldots,z_k)$ with integer coefficients such that the
1 KB (165 words) - 16:59, 14 February 2020
where the polynomial $ f( x) = x ^ {n} + a _ {1} x ^ {n-} 1 + \dots + a _ {n} $
is not the square of another polynomial modulo $ p $,
7 KB (1,033 words) - 17:46, 4 June 2020
...; x)$ of order $n$ of $f$ is the [[Trigonometric polynomial|trigonometric polynomial]]
...or $S_\alpha(\vb{x})$ is always a [[Trigonometric polynomial|trigonometric polynomial]].
6 KB (1,047 words) - 11:37, 13 February 2024
is a Laguerre polynomial, is sometimes also called a Laguerre function.
1 KB (164 words) - 22:15, 5 June 2020
...ldots,x_n\}$ are the vertices of $\Delta$, $k[x_1,\ldots,x_n]$ denotes the polynomial ring over $k$ in the variables $\{x_1,\ldots,x_n\}$, and $I_\Delta$ is the
...ner ring, called the exterior face ring or indicator algebra, in which the polynomial ring $k[x_1,\ldots,x_n]$ in the definition of $k[\Delta]$ is replaced by th
3 KB (544 words) - 20:16, 8 November 2023
A form in two variables, viz. a homogeneous [[Polynomial|polynomial]]
4 KB (512 words) - 10:59, 29 May 2020
The Dickson polynomial of the first kind of degree $n$ with parameter $a$ is defined by
...ssical [[Chebyshev polynomials|Chebyshev polynomials]]. In particular, the polynomial $D _ { n } ( x , a )$ satisfies
15 KB (2,207 words) - 16:45, 1 July 2020
...The main link invariant yielded by a Conway algebra is the [[Jones–Conway polynomial]], [[#References|[a2]]], [[#References|[a5]]], [[#References|[a4]]].
...</td> <td valign="top"> A.S. Sikora, "On Conway algebras and the Homflypt polynomial" ''J. Knot Th. Ramifications'' , '''6''' : 6 (1997) pp. 879–893</td><
8 KB (971 words) - 07:05, 24 March 2024
...in the components of the curvature tensor of $V^{2p}$, and $\phi$ is some polynomial in the components of the curvature tensor and the coefficients of the secon
4 KB (553 words) - 09:38, 22 August 2014
$f$ is a non-zero polynomial with coefficients from $k$ (this
polynomial may be taken to be irreducible). The smallest extension of
11 KB (1,965 words) - 04:47, 16 January 2022
The number of spanning trees is a special value of the [[Tutte polynomial]] of the graph, $T_G(1,1)$.
1 KB (198 words) - 12:59, 29 December 2014
or over any local ring is local. On the other hand, the polynomial ring $ k [ X _ {1} \dots X _ {n} ] $
this function coincides with a certain polynomial $ \overline{H}\; _ {A} ( n) $
9 KB (1,456 words) - 22:17, 5 June 2020
...lgebra]]. As a consequence of the lower bound on the degree of the minimal polynomial of the Grasmann derivative, the following theorem is true [[#References|[a3
...op">[a2]</TD> <TD valign="top"> N. Alon, M.B. Nathanson, I. Z. Rusza, "The polynomial method and restricted sums of congruence classes" ''J. Number Th.'' 56, N
4 KB (577 words) - 10:26, 10 December 2023
...e strictly less than one in modulus. In the case given, the characteristic polynomial $ \rho ( \lambda ) = \lambda ^ {2} - 1 $
4 KB (538 words) - 17:45, 20 January 2022
...a natural number, the Hermite equation has among its solutions the Hermite polynomial of degree $n$ (cf. [[Hermite polynomials|Hermite polynomials]]),
1 KB (190 words) - 19:23, 16 October 2023
The nodes $x_j$ are the roots of the Jacobi polynomial $P_n^{(0,1)}(x)$ (the [[Jacobi polynomials|Jacobi polynomials]] form an ort
1 KB (211 words) - 12:26, 27 October 2014
...ords of a given word length, those which satisfy a predicate computable in polynomial time. One can also mention the complexity of the computations performed by
...n simulate each other with a constant multiplicative factor in space and a polynomial time overhead. The latter property characterizes all "reasonable" models
16 KB (2,413 words) - 16:10, 1 April 2020
A [[Field|field]] $k$ over which every polynomial is separable. In other words, every algebraic extension of $k$ is a [[Separ
1 KB (231 words) - 21:49, 11 October 2014
...ves of a function by means of the derivative of the Lagrange interpolating polynomial (cf. also [[Lagrange interpolation formula]]). For given interpolation node
For $x _ { 1 } , \ldots , x _ { m }$ being the zeros of the Jacobi polynomial (cf. [[Jacobi polynomials|Jacobi polynomials]]) $P _ { m } ^{( \alpha , \be
6 KB (756 words) - 20:24, 5 December 2023
other than a polynomial, there exists a comparison function $ A ( z) $,
...TR><TD valign="top">[1]</TD> <TD valign="top"> R.P. Boas, R.C. Buck, "Polynomial expansions of analytic functions" , Springer & Acad. Press (U.S.A. &
5 KB (701 words) - 20:12, 10 January 2021
where $a_0$ denotes the leading coefficient and $N$ the degree of the minimal polynomial $f$ (with integral coefficients) of $\alpha$ (cf. also [[Algebraic number]]
...lue for $M(\alpha)$ equals $1.32471\ldots$ and is attained by roots of the polynomial $X^3-X-1$.
7 KB (1,029 words) - 07:50, 27 March 2018
There is a polynomial time algorithm for computing the Dilworth number of a finite graph.
1 KB (191 words) - 19:14, 14 November 2023
...gonometric and Gamma-functions, Hermite polynomials) and (differential and polynomial) equation solving.
...ith ideals and modules generated by polynomials or polynomial vectors over polynomial or power series rings and generalizations).
9 KB (1,292 words) - 20:51, 18 September 2016
If $R ( z )$ is a polynomial of degree $m$, then
...{ n } , 1 / z _ { 1 } , \dots , 1 / z _ { n }$ by associating to any such polynomial its free term (L. Aizenberg, cf. [[#References|[a1]]]).
7 KB (1,090 words) - 10:25, 16 March 2024
...on $f=0$, respectively the inequality $f\ne 0$, to it ($f$ is an arbitrary polynomial); or, in one of the systems, two inequalities are replaced by their product
...ty, $f\ne 0$, the inequality $f_0\ne 0$ is contained in it, where $f_0$, a polynomial in $x_2,\dots,x_n$, is the leading coefficient of the expansion of $f$ in p
8 KB (1,363 words) - 16:04, 13 January 2021
...ntial, although the algorithmic complexity of the class as a whole is only polynomial), until now (1990) no serious competitors have been suggested.
...[[#References|[a3]]]. Alternative algorithms for linear programming with a polynomial-time worst-case behaviour have been proposed by L.G. Khachiyan [[#Reference
8 KB (1,246 words) - 06:35, 7 May 2022
...mials]]), where $E _ { n + 1 } ( x ) = T _ { n + 1 } ( x )$, the Chebyshev polynomial of the first kind. For $h ( x ) = ( 1 - x ^ { 2 } ) ^ { - 1 / 2 }$, $P _ {
where $\rho _ { m }$ is a polynomial of degree $m$ that is positive in $[ - 1,1 ]$ [[#References|[a6]]], [[#Refe
7 KB (1,048 words) - 20:47, 11 February 2024
...hand sides it follows from Hilbert's theorem on the finiteness of bases of polynomial ideals that in this sequence only finitely many are essential and that the
4 KB (599 words) - 15:17, 14 February 2020
$#C+1 = 35 : ~/encyclopedia/old_files/data/N066/N.0606230 Negative polynomial distribution,
The [[Generating function|generating function]] of the negative polynomial distribution with parameters $ r, p _ {0}, \dots, p _ {k} $
3 KB (418 words) - 16:50, 1 February 2022
A theorem which in some cases enables one to give effectively the polynomial and the error of best integral approximation of a function $ f $.
If the polynomial
3 KB (440 words) - 12:21, 18 February 2022
...]]], G. Hermann set out to show that all standard objects in the theory of polynomial ideals over fields $k$, including the prime ideals associated to a given id
...that it is necessary to assume that one can completely factor an arbitrary polynomial over $k$.
10 KB (1,498 words) - 14:47, 3 February 2024
amounts theoretically to constructing the characteristic polynomial $ \phi _ {A} $
i.e. its multiplicity as a root of the characteristic polynomial $ \phi _ {A} $)
15 KB (2,340 words) - 08:54, 21 January 2024
is orthogonal to every trigonometric polynomial of order not exceeding $ n - 1 $.
1 KB (199 words) - 19:36, 2 January 2021
...o [[Extension of a field|Extension of a field]]). Then the monic separable polynomial whose roots are precisely the elements of $V$ has the form
<tr><td valign="top">[a2]</td> <td valign="top"> D. Benson, "Polynomial invariants of finite groups" , ''London Math. Soc. Lecture Notes'' , '''190
6 KB (860 words) - 16:28, 12 July 2020
...regression is far from being linear, one can pose the problem of finding a polynomial $ g ( x) = \beta _ {0} + \beta _ {1} x + \dots + \beta _ {m} x ^ {m} $
A solution of this problem corresponds to polynomial mean-square regression (see [[Parabolic regression|Parabolic regression]]).
10 KB (1,524 words) - 08:10, 6 June 2020
by successive squaring operations in order to reduce the degree of the polynomial $ f( x) $.
is a polynomial of degree $ n - 1 $
5 KB (759 words) - 08:29, 6 June 2020
is a homogeneous polynomial with algebraic coefficients, $ k $
is a fixed non-zero algebraic number, and the polynomial $ F ( X,1 ) $
7 KB (1,087 words) - 19:41, 5 June 2020
...the number $n$ is called the order of the curve $L$. If $f$ is a reducible polynomial, i.e. if it splits into factors $f_1, \ldots, f_k$, then the curve $L$ defi
If on the other hand $f$ is an irreducible polynomial, $L$ is called an irreducible curve. Two irreducible plane real algebraic c
11 KB (1,916 words) - 00:44, 12 August 2019
A greatest common divisor of two polynomials over a given field is a polynomial of greatest degree that divides both polynomials. In this case again, a gr
...Examples of such domains are the ring $\mathbb{Z}$ of rational integers or polynomial rings $F[X]$, where $F$ is a [[field]] (e.g., $\mathbb{C}$ or $\mathbb{R}$
4 KB (673 words) - 17:01, 26 October 2014
...x_k \in [a,b]$, $k=0,\ldots,n$, are given, then the Lagrange interpolation polynomial $L_n(x)$ (see [[Lagrange interpolation formula]]), which is defined on the
2 KB (274 words) - 21:34, 22 April 2017
v) $d ^ { * }$ is a polynomial in $\sum _ { j = 1 } ^ { n } x _ { j }$, $x _ { j } \in \{ 0,1 \}$, $j = 1
...{ n } } ( d ^ { * } )$, $n \in \mathbf N$, if and only if $d ^ { * }$ is a polynomial in $k \in \mathbf{N} \cup \{ 0 \}$ of degree not exceeding $\delta$. In par
13 KB (1,906 words) - 17:44, 1 July 2020
then $P_n(x)$ is the polynomial of best uniform approximation for $f(x)$, i.e.
3 KB (403 words) - 17:05, 28 June 2015
...an the characteristic properties of the [[Polynomial of best approximation|polynomial of best approximation]] (see [[#References|[8]]]). However, in $ L [ a ,
is a polynomial spline of degree $ m $
13 KB (1,959 words) - 08:22, 6 June 2020
...responds to the given ideal. As distinct from Hilbert's basis theorem in a polynomial ring, the essential feature of the Ritt–Raudenbush theorem is that the id
...ce over some universal extension of the field of coefficients on which any polynomial from this ideal vanishes. The analogue of Hilbert's Nullstellensatz is vali
30 KB (4,468 words) - 18:44, 17 December 2019
be a homogeneous polynomial of degree $ 2m $
As a polynomial in $ \tau $,
8 KB (1,200 words) - 09:02, 1 February 2020
====Discriminant of a [[polynomial]]====
The discriminant of a polynomial $f(x)=a_0 x^n+a_1 x^{n-1}+\cdots + a_n$, $a_0 \ne 0$, whose roots are $\def
16 KB (2,947 words) - 08:53, 9 December 2016
is the degree of the polynomial $ Q ( z) $
2 KB (222 words) - 12:51, 6 January 2024
...w and differential polynomial rings are a special case of the general-skew-polynomial ring $ F [ x; S, d] $,
5 KB (880 words) - 19:00, 9 January 2024
...{ 2 } } ( L _ { n } )$ is bounded by a polynomial for any $L$ decidable in polynomial time (cf. [[#References|[a10]]] and also [[NP|$\cal N P$]]).
...r><td valign="top">[a15]</td> <td valign="top"> Y. Yao, "Separating the polynomial-time hierarchy by oracles" , ''Proc. 26th Ann. IEEE Symp. Found. Comput. Sc
9 KB (1,354 words) - 17:44, 1 July 2020
...$k \le n$ and $C_k = 0$ for $k > n$. This defines an imbedding $i$ of the polynomial ring $A[T_1,\ldots,T_N]$ into $A[[T_1,\ldots,T_N]]$. There is a topology de
...Then $F = UP$, where $U$ is an invertible formal power series and $P$ is a polynomial of the form $T^k + a_{k-1}T^{k-1} + \cdots + a_0$, where the coefficients $
6 KB (1,093 words) - 08:26, 16 March 2023
...ith integer coefficients has integer values of the variables that make the polynomial vanish. Many decidability problems did not yield a solution for a long time
4 KB (655 words) - 17:22, 7 February 2011
...ss, Carl Friedrich|C.F. Gauss]] it is known that an arbitrary [[Polynomial|polynomial]] over a [[Field|field]] or over the integers can be factored into irreduci
...d (J. von zur Gathen and V. Shoup [[#References|[a8]]]) uses the so-called polynomial representation of $\sigma$ as its basic tool. The last two algorithms are b
12 KB (1,739 words) - 13:11, 26 March 2023
where $f(x,y,z)=0$ is a homogeneous polynomial of degree $n$ (a form of degree $n$ in $x,y,z$). If the point $M(x_0,y_0,z_
1 KB (217 words) - 17:29, 11 April 2014
...mined from the condition that formula \eqref{eq:1} be exact if $f(x)$ is a polynomial of degree at most $n$.
2 KB (240 words) - 21:28, 18 January 2018
...group (cf. also [[Polynomial and exponential growth in groups and algebras|Polynomial and exponential growth in groups and algebras]]; [[#References|[a1]]], [[#R
4 KB (659 words) - 08:29, 6 June 2020
and if $A$ is invertible, then $A^{-k} = (A^{-1})^k$. For the polynomial
matrix polynomial
18 KB (3,377 words) - 17:54, 2 November 2013
...ield $\mathbf Q$ of rational numbers: every $\alpha\in K$ is the root of a polynomial (of degree at most $n$) over $\mathbf Q$. A number field that is not algeb
2 KB (261 words) - 20:42, 23 November 2023
Given a [[Trigonometric polynomial|trigonometric polynomial]]
The Carathéodory–Toeplitz extension problem for the trigonometric polynomial (a4) is solvable if and only if the matrix
41 KB (5,422 words) - 22:26, 1 January 2018
...m (LS) near a Fuchsian singular point $t=t_*$ can be always brought into a polynomial [[normal form]]. To describe this form, assume for simplicity that $t_*=0$
...normal form]] (upper triangular). In the resonant case the normal form is polynomial,
7 KB (1,237 words) - 11:48, 23 November 2023
There is a polynomial time algorithm for computing the Dilworth number of a finite graph and so f
1 KB (209 words) - 18:14, 1 June 2023
In other words, the polynomial $ T _ {n} $
is a polynomial of degree $ \leq n $)
5 KB (721 words) - 19:38, 5 June 2020
The characteristic polynomial of $ {\mathcal A} $
...anifold]]). Its [[Cohomology|cohomology]] is torsion-free and its Poincaré polynomial (cf. [[Künneth formula|Künneth formula]]) is $ { \mathop{\rm Poin} } (
9 KB (1,307 words) - 18:48, 5 April 2020
...is ultimately periodic with period $r$ if and only if $(1-x^r)a(x)$ is a [[polynomial]] over $F$. Any ultimately periodic sequence over a field is a [[shift regi
2 KB (266 words) - 15:39, 24 September 2016
...form $\alpha(x)=p(x)/q(x)$, with $p(x)=1-p_1x-\dotsb-p_mx^m$ and $q(x)$ a polynomial of degree $\leq m-1$.
1 KB (226 words) - 13:04, 14 February 2020
the Brown–Peterson spectrum. The homotopy of this spectrum is the polynomial algebra
...us [[#References|[a3]]] and M. Hazewinkel [[#References|[a2]]] constructed polynomial generators of $ \pi _ {*} BP $
4 KB (633 words) - 08:47, 26 March 2023
be the interpolation polynomial of degree $ \leq n $
th Chebyshev polynomial, the Lebesgue constants have minimum order of growth, namely
7 KB (982 words) - 20:31, 15 November 2023
...mming|linear programming]] and [[#References|[a7]]] for solving systems of polynomial equations. For problems with inputs from infinite-dimensional spaces, often
...ies of $\mu$, the average cost of the [[Simplex method|simplex method]] is polynomial, while the worst-case (maximal) cost is exponential, see [[#References|[a1]
6 KB (908 words) - 18:44, 21 March 2024
A polynomial in several variables all terms of which are of the same degree.
1 KB (237 words) - 07:31, 19 January 2016
$#C+1 = 209 : ~/encyclopedia/old_files/data/P073/P.0703700 Polynomial and exponential growth in groups and algebras
is of polynomial growth, or power growth, $ r $
19 KB (2,908 words) - 20:20, 12 January 2024
is a polynomial in the variables $ x _ {1} \dots x _ {n} $
The study of the number of solutions of congruences in the case of a general polynomial $ F( x) $
8 KB (1,148 words) - 17:46, 4 June 2020
...and several other radial basis functions, a linear (generally, low-order) polynomial has to be added to $s$ with side conditions $\sum_{j=1}^m\lambda_j=\sum_{j=
...rpolation (cf. e.g. [[Algebraic polynomial of best approximation|Algebraic polynomial of best approximation]]), are often ruled out.
8 KB (1,246 words) - 15:13, 18 February 2013
...ho also proposed the following method of constructing a Sturm series for a polynomial $ f ( x) $
...ng to a real-closed field. The algorithm to determine a Sturm series for a polynomial $ f _ {0} ( x) $
3 KB (516 words) - 10:42, 19 February 2021
Given inequalities of the type $\|z^{n+1}\|\leq C\|z^n\|^k$, one speaks of a polynomial rate of convergence of order $k$ (for example, the quadratic rate of conver
2 KB (287 words) - 12:05, 13 August 2014
is the characteristic polynomial of the isometry $ T $.
A polynomial $ \lambda ( t) = t ^ {k} + a _ {1} t ^ {k - 1 } + \dots + 1 $
11 KB (1,631 words) - 17:45, 4 June 2020
be the characteristic polynomial of $A$, where $ { I } _ { n }$ is the $( n \times n )$ identity matrix. The
be the matrix characteristic polynomial and let $\Delta \in C ^ { n \times n }$ be the matrix (block) eigenvalue of
15 KB (2,237 words) - 17:01, 1 July 2020
..., see [[Galois theory|Galois theory]] and [[Primitive polynomial|Primitive polynomial]].
The standard alternative to using a polynomial basis is a normal basis, that is, a basis of the form $\{ \alpha , \alpha ^
14 KB (2,032 words) - 14:35, 19 March 2023
Pick an irreducible monic primitive [[Polynomial|polynomial]] $ {\overline{f}\; } $
and lift it to a polynomial $ f $
7 KB (1,070 words) - 19:40, 5 June 2020
...ise to invariants for knots and links in $3$-manifolds including the Jones polynomial for knots in the $3$-sphere. See also [[#References|[a1]]] and [[#Reference
...]</td> <td valign="top"> E. Witten, "Quantum field theory and the Jones polynomial" ''Comm. Math. Phys.'' , '''121''' (1989) pp. 351–399</td></tr></table
4 KB (628 words) - 16:58, 1 July 2020
...the linear complexity $L ( \mathbf{a} )$ is just the degree of its minimal polynomial $m$, i.e. the length of a shortest linear feedback shift register (LFSR; cf
4 KB (622 words) - 19:04, 17 February 2024
the former definition means that for any unitary polynomial $ P( X) \in A[ X] $
For a discussion of the solution of systems of polynomial equations and implicit-function type statements in the context of Hensel ri
5 KB (787 words) - 22:10, 5 June 2020
...Jacobson radical are now called semi-primitive rings. Primitive rings with polynomial identities are central simple finite-dimensional algebras. Primitive rings
2 KB (275 words) - 18:10, 11 April 2014
...os x)$, which are related to Chebyshev polynomials of the second kind. The polynomial $T_n(x)$ is a real solution to Chebyshev's equation on the entire real line
1 KB (219 words) - 19:57, 21 November 2018
...z$ and $w$ which are related by the equation $w^2=F(z)$, where $F(z)$ is a polynomial of degree 3 or 4 without multiple roots. The complete solution of this prob
2 KB (250 words) - 22:06, 28 November 2014
be the interpolation polynomial for $ f ( x) $
is the Lagrangian basis polynomial of the $ i $-th node (cf. [[Lagrange interpolation formula|Lagrange inter
11 KB (1,719 words) - 01:56, 4 January 2022
The Laurent polynomial $ \Delta _ {i} (t _ {1} \dots t _ \mu ) $
is simply called the Alexander polynomial of $ k $(
12 KB (1,694 words) - 06:42, 26 March 2023
The Pfaffian (of a [[skew-symmetric matrix]] $X$) is the polynomial $\def\Pf{\mathrm{Pf}\;} \Pf X$ in the entries of $X$ whose square is the
1 KB (251 words) - 19:57, 30 November 2014
are of minimal order. In order that a polynomial $ P ( \lambda ) $
...ecessary and sufficient that it can be divided by the minimal annihilating polynomial of the operator $ U _ {\mathcal T} $.
9 KB (1,299 words) - 08:07, 6 June 2020
satisfies the following condition: The roots of the polynomial
2 KB (245 words) - 17:32, 5 June 2020
...y class]] $\mathrm{BPP}$ (an abbreviation of bounded-error, probabilistic, polynomial-time) is formally defined as
This means it holds $L\in \mathrm{BPP}$ if a polynomial-time PTM $M$ exists with
12 KB (1,954 words) - 17:47, 26 December 2013
...$\mathbf{Z}\langle X \rangle$ becomes the [[Leibniz–Hopf algebra]]. A Lie polynomial is an element $P$ of $\mathbf{Z}\langle X \rangle$ such that $\mu(P) = 1 \o
2 KB (298 words) - 20:59, 9 December 2015
...ween polynomial and exponential; however, a [[regular language]] is either polynomial or exponential.
6 KB (991 words) - 07:57, 18 November 2023
that is, one may associate a [[Lie polynomial|Lie polynomial]] $ \psi ( t ) $
5 KB (722 words) - 18:41, 5 October 2023
contains all roots of the minimum polynomial of $ k $ .
...l form, or when the ground field does not contain all roots of the minimum polynomial of the matrix (see [[#References|[2]]]–[[#References|[5]]]).
5 KB (708 words) - 11:11, 17 December 2019
...anet is expressed approximately as the argument of a certain trigonometric polynomial
...ferences|[1]]]–[[#References|[3]]]). The final result that a trigonometric polynomial always has an average rotation was stated in 1938 by B. Jessen (for its pro
5 KB (699 words) - 11:07, 25 April 2020
...w.encyclopediaofmath.org/legacyimages/o/o070/o070660/o0706609.png" /> is a polynomial in one variable (an ordinary differential equation) or several variables (a
5 KB (715 words) - 08:35, 19 October 2014
...ion that the function $[f(\lambda)]^{-1}$ is the square of a trigonometric polynomial of fixed order. A more general class of parametric spectral estimators ofte
2 KB (304 words) - 13:00, 16 July 2014
Jackson's singular integral is a positive operator and is a trigonometric polynomial of order $ 2n - 2 $;
2 KB (252 words) - 22:14, 5 June 2020
The Szegö polynomials form an orthogonal polynomial sequence with respect to the positive definite Hermitian [[Inner product|in
...the open unit disc) satisfying $H ( 0 ) = 1$. If $H$ is restricted to be a polynomial of degree at most $n$, then a solution is given by $H = \Phi _ { n } ^ { *
7 KB (1,105 words) - 10:02, 11 November 2023
of the polynomial
2 KB (258 words) - 17:32, 5 June 2020
...on of an arbitrary positive integral power of a [[Binomial|binomial]] in a polynomial arranged in powers of one of the terms of the binomial:
2 KB (262 words) - 08:02, 6 June 2020
...point" or "Chebyshev node" is also used to denote a zero of a Chebyshev polynomial (cf. [[Chebyshev polynomials|Chebyshev polynomials]]) in the theory of (num
2 KB (296 words) - 16:59, 22 November 2018
A lemniscate is a level curve of a polynomial. If all the foci $F_k$: $z_k=x_k+iy_k$, $k=1,\dotsc,n$, are distinct and th
2 KB (304 words) - 06:49, 7 April 2023
[[Irreducible polynomial|Irreducible polynomial]]). The formula
The Lucas polynomials $V_n(x)$ are a special case of the [[Dickson polynomial]]s $D_n(x,a)$ by taking $a$ to be $-1$.
7 KB (1,115 words) - 16:05, 22 January 2021
One can take spaces of polynomial, rational, real-analytic, smooth, or $L _ { 1 / 2 } ^ { 2 }$ loops, in decr
...h loop group is an infinite-dimensional Fréchet manifold, the rational and polynomial loop groups have natural filtrations by finite-dimensional algebraic subvar
5 KB (698 words) - 08:13, 15 February 2024
...is to be interpreted as $-g(\gamma)^{-1} f(z)$ where $f(z)$ is the unique polynomial of degree ${}< \deg g(z)$ such that $(z-\gamma) f(z) \equiv 1 \pmod{g(z)}$.
5 KB (779 words) - 07:49, 26 January 2018
for any polynomial $p(z)$. Thus, the unit circle is a spectral set for any contraction (an ope
2 KB (295 words) - 15:46, 29 December 2018
...he class of all algebras satisfying a given system of identical relations (polynomial identities in the case of rings and algebras over a field). The description
...ebra|Lie algebra]] of all $(2\times2)$-matrices has no finite basis of its polynomial identities. Recently (1999), A.Ya. Belov [[#References|[a3]]], see also [[#
7 KB (1,092 words) - 13:48, 14 February 2020
An algebra over a field for which certain polynomial identities are true.
is said to be a polynomial identity of the algebra $ A $
15 KB (2,252 words) - 08:04, 6 June 2020
...approximation theory. E.g., using it one can prove that there is a unique polynomial of degree $ n $
2 KB (342 words) - 08:27, 6 June 2020
...$ into $k$). If $f(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_0$ is the characteristic polynomial of $\a\in K$ relative to $k$, then
2 KB (317 words) - 08:59, 15 April 2012
...for Hilbert's 16th problem (i.e., what is the number of limit cycles of a polynomial vector field in the plane?) see [[#References|[a2]]]. Well-known is the [[S
...><TD valign="top">[a2]</TD> <TD valign="top"> N.G. Lloyd, "Limit cycles of polynomial systems - some recent developments" T. Bedford (ed.) J. Swift (ed.) , ''New
5 KB (770 words) - 18:12, 16 December 2020
and the Chern polynomial is the expression $ c _ {t} = 1 + c _ {1} t + c _ {2} t ^ {2} + \dots $,
and the Chern polynomial as an element of the formal power series ring $ H ^ {**} ( \mathop{\rm
13 KB (1,910 words) - 19:40, 7 January 2024
is a polynomial of degree not exceeding $ \mu $.
is not a polynomial, then $ M ( r) $
8 KB (1,251 words) - 20:13, 10 January 2021
is identically zero for some [[Polynomial|polynomial]] $ A ( x,y,z ) $.
5 KB (746 words) - 06:34, 26 March 2023
The equation $F=0$, when $F$ is a polynomial in $y'$, is also studied in the complex domain [[#References|[2]]].
3 KB (417 words) - 11:55, 26 March 2023
whereby the degree of every polynomial $ P _ {n} $
if every polynomial has positive leading coefficient and if the normalizing condition
20 KB (2,878 words) - 11:11, 4 January 2021
...olynomials over the field of complex numbers which contains many classical polynomial systems. The Appell polynomials were introduced by P.E. Appell [[#Reference
is tantamount to saying that the degree of the polynomial $ {A _ {n} } (z) $
11 KB (1,595 words) - 17:13, 2 January 2021
The modern umbral calculus is designed to study polynomial sequences $p_n(x)$ ''of binomial type'', that is, sequences for which $\deg
as well as polynomial sequences $s_n(x)$ ''of Sheffer type'', that is, sequences for which $\deg
7 KB (1,142 words) - 20:34, 9 December 2015
...terpolation formula]], one constructs the (unique) algebraic interpolation polynomial $ p _ {n} ( z) $
...ase in the presence of multiple nodes the (unique) algebraic interpolation polynomial $ p _ {n} ( z) $
12 KB (1,759 words) - 19:07, 11 January 2024
where $H_\nu(x)$ is the Hermite polynomial (cf. [[Hermite polynomials|Hermite polynomials]]) [[#References|[2]]], [[#R
2 KB (288 words) - 20:16, 28 January 2020
...#References|[a6]]] as a tool to study prime rings satisfying a generalized polynomial identity. Specifically, let $R$ be a [[Prime ring|prime ring]] (with $1$) a
...s|[a6]]] asserts that a prime ring $R$ satisfies a non-trivial generalized polynomial identity if and only if $R C$ has an idempotent $e$ such that $e R C$ is a
10 KB (1,646 words) - 19:47, 23 December 2023
Gregory's formula is obtained by the integration of the interpolation polynomial with nodes at $ a, a + h \dots a + nh $.
...of the [[Fraser diagram|Fraser diagram]] and by integrating the resulting polynomial interpolation formula in each interval $ [ a + jh , a + jh + h ] $.
6 KB (781 words) - 19:42, 5 June 2020
...h algebraic coefficients. Assuming $\alpha^\beta$ is algebraic, a non-zero polynomial $P$ is constructed so that $F$ vanishes at many such points $u + v\beta$; t
5 KB (726 words) - 09:23, 20 December 2014
be a polynomial of degree at most $ n - 1 $.
...sc, that is, the Schur algorithm can be used to determine whether or not a polynomial $ p ( z ) $
6 KB (836 words) - 11:17, 30 May 2020
if for each polynomial $ f (x _{1} \dots x _{n} ) $
...braically independent over the field of algebraic numbers, i.e. if for any polynomial $ P (x _{1} \dots x _{n} ) $
6 KB (793 words) - 17:24, 17 December 2019
...ring of the polynomial ring $\CH(X)[\zeta]$ by the ideal generated by the polynomial
4 KB (714 words) - 21:54, 24 April 2012
can be computed recursively by dividing the polynomial $ x ^ {n} - 1 $
The polynomial $ \Phi _ {n} ( x) $
4 KB (652 words) - 05:18, 7 March 2022
...s consisting of nilpotent elements is nilpotent. If an algebra satisfies a polynomial identity of degree $d$, then every nilpotent subring of it of degree $[d/2]
2 KB (370 words) - 13:19, 10 April 2014
have the same characteristic polynomial, determinant, trace, and eigen values. In a unitary space, their characteri
2 KB (314 words) - 16:09, 1 April 2020
...s outside (a countable intersection of) open dense set(s). For instance, a polynomial has generically no double roots. Two submanifolds $A,B\subset N$ are in gen
2 KB (324 words) - 12:46, 4 September 2014
is sought in the form of an interpolation polynomial in $ h $.
the method reduces to the interpolation of a polynomial in $ h ^ {p} $,
7 KB (1,037 words) - 17:08, 13 January 2024
...ucted (cf. also [[Polynomial and exponential growth in groups and algebras|Polynomial and exponential growth in groups and algebras]]).
8 KB (1,322 words) - 13:08, 10 February 2024
...Movable singular point]]) and transcendental branch points, where $P$ is a polynomial in the unknown function $w$ and its derivative $w'$, while $P$ is an analyt
2 KB (326 words) - 14:46, 17 July 2014
...R><TR><TD valign="top">[6]</TD> <TD valign="top"> M. Gromov, "Groups of polynomial growth and expanding maps" ''Publ. Math. IHES'' , '''53''' (1981) pp. 53
2 KB (337 words) - 19:13, 9 October 2014
...open question if the linear programming problem is solvable by a strongly polynomial algorithm, the running time of which is polynomially bounded in the dimensi
...atural probabilistic models, several variants of the simplex method run in polynomial expected time.
12 KB (1,882 words) - 22:25, 1 September 2016
is a polynomial with leading coefficient one, then this solution is called a Lamé function
2 KB (339 words) - 22:15, 5 June 2020
2 KB (354 words) - 19:08, 27 December 2014
...raically closed; or 3) every positive element in $k$ is a square and every polynomial of odd degree over $k$ has a root in $k$. Every formally-real field has a r
2 KB (375 words) - 19:58, 7 December 2014
A form of writing the polynomial $ H _ {m} $
3 KB (522 words) - 20:12, 10 January 2024
is an infinite cyclic covering of the complement of the knot. The polynomial matrix $ t A + ( - 1 ) ^ {q} A ^t $
<TR><TD valign="top">[3]</TD> <TD valign="top"> J. Levine, "Polynomial invariants of knots of codimension two" ''Ann. of Math.'' , '''84''' (196
5 KB (839 words) - 06:50, 28 April 2024
...unctional. A positive distribution can only exist if $L ( p ) > 0$ for any polynomial $p$ that is positive on $I$.
...functional $L$ is positive if $L ( p ^ { 2 } ( x ) ) > 0$ for any non-zero polynomial $p$ and this implies that the moment matrices, i.e., the Hankel matrices of
6 KB (888 words) - 18:32, 21 December 2020
is a polynomial of degree not exceeding $ m $,
2 KB (385 words) - 16:12, 1 August 2021
...{ m - 1 }$. Assume that $p > N$ and $m = 2$ and let $q_{l}$ be a piecewise polynomial of degree $1$ which interpolates the values of the function $u$ at the vert
...bounded in $W _ { p } ^ { m } ( T )$. Obviously, $F ( u ) = 0$ if $u$ is a polynomial of degree $1$, since the linear interpolant recovers $u$ exactly. Therefore
11 KB (1,716 words) - 17:52, 3 January 2021
...plied to determine values at $2 N$ equi-spaced points of the trigonometric polynomial that approximates $y = K x$ (cf. also [[Fourier series|Fourier series]]). B
7 KB (1,076 words) - 07:40, 7 February 2024
A form of notation of the interpolation polynomial obtained from the [[Stirling interpolation formula|Stirling interpolation f
3 KB (346 words) - 08:23, 6 June 2020
...nd and older results. The paper [[#References|[a24]]] is an early study on polynomial convexity.
Polynomial convexity arises naturally in the context of function algebras (cf. also [[
20 KB (3,071 words) - 17:45, 1 July 2020
...he Steiner formula holds: The volume of the point set bounded by them is a polynomial of degree 3 in $\epsilon$. An analogous result holds for arbitrary dimensio
2 KB (319 words) - 17:16, 3 August 2014
...s clear from this equation that $s_{ \lambda }$ is a symmetric homogeneous polynomial of degree $| \lambda | = \Sigma _ { i } \lambda_i$.
...004094.png"/> is a class function). In fact, more is true: The irreducible polynomial characters of $\operatorname{GL}_l$ are precisely the $s_{ \lambda }$ for $
14 KB (2,001 words) - 10:09, 11 November 2023
be a polynomial of degree $ N $
The Newton polygon of the polynomial $ f( X) $
7 KB (1,067 words) - 12:49, 12 March 2021
is a polynomial), and similarly for $ B ( \xi ) $.
2 KB (356 words) - 08:07, 6 June 2020
...to solve it, cf. [[#References|[a10]]]. Propositional Horn clauses have a polynomial-time solvable satisfiability problem. In fact, linear time solutions have b
...rization of classes of finite ordered structures which are recognizable in polynomial time. Other characterizations of these model classes in terms of fixed-poin
7 KB (1,111 words) - 06:59, 21 October 2016
Let $V$ be finite-dimensional. If all roots of the [[characteristic polynomial]] of $A$ are in $K$ (e.g. if $K$ is algebraically closed), then $V$ decompo
2 KB (367 words) - 19:45, 16 November 2017
...$ which is a simple root of the [[Characteristic polynomial|characteristic polynomial]]. For any eigenvalue $\lambda$ of $L$, $r \geq | \lambda |$; indeed $L$ ha
6 KB (894 words) - 20:14, 4 February 2024
are polynomials, the analytic polyhedron is said to be a polynomial polyhedron. If $ m = n $
2 KB (335 words) - 18:47, 5 April 2020
...al perturbation term can be used to produce a truncated series, that is, a polynomial solution.
...m approximation]] of zero over the same interval $J$, which is a Chebyshev polynomial $T _ { n } ^ { * } ( x )$ of degree $n$, defined over $J$ (cf. also [[Cheby
20 KB (2,974 words) - 16:52, 1 July 2020
...timated intercept, $\hat{\beta}_0$, in the weighted least squares fit of a polynomial of degree $p$,
...oted that the Nadaraya-Watson estimator (1) is a special case of the local polynomial regression estimator with $p=0$. In practice, the local linear ($p=1$) and
14 KB (2,191 words) - 10:16, 4 April 2016
...n$ that decrease at infinity, as do all their derivatives, faster than any polynomial, with the topology given by the system of semi-norms
2 KB (359 words) - 22:13, 26 July 2012
that is, a [[Polynomial|polynomial]], is also called an entire rational function. Otherwise it is called a fra
is a polynomial, called the integral part of the fraction $ P / Q $,
8 KB (1,257 words) - 03:49, 4 March 2022
...al degree of the Jones polynomial, and to half the degree of the Alexander polynomial.
6 KB (918 words) - 16:58, 1 July 2020
...l E$ of a vessel $\frak V$ as in (a1) is finite dimensional; one defines a polynomial in two complex variables $\lambda _ { 1 }$, $\lambda _ { 2 }$ by setting
...{E}$. One calls $p ( \lambda _ { 1 } , \lambda _ { 2 } )$ the discriminant polynomial of the vessel $\frak V$, and the projective [[Plane real algebraic curve|pl
24 KB (3,136 words) - 20:00, 24 November 2023
...align="top">[a1]</TD> <TD valign="top"> R.P. Feinerman, D.J. Newman, "Polynomial approximation" , Williams & Wilkins pp. Chapt. IV.4</TD></TR></table>
2 KB (339 words) - 19:38, 5 June 2020
The non-zero roots of the [[characteristic polynomial]] of a real skew-symmetric matrix are purely [[imaginary number]]s. A real
2 KB (339 words) - 06:23, 12 September 2016
The ultraspherical polynomial $ C _ {n} ^ {( \lambda ) } ( x) $
3 KB (417 words) - 07:38, 26 February 2022
is a polynomial of degree $ n+ 1 $
it is usually called the Neumann polynomial of order $ n $.
6 KB (896 words) - 19:13, 17 January 2024
...each point $a\in U$ and each derivative $f^{(\a)}$ one can form the Taylor polynomial of order $r\le m-|\a|$ centered at $a$,
The difference between $f^{(\a)}(x)$ and the value provided by the Taylor polynomial $\Big(T_a^r f^{(\a)}\Big)(x)$ should be small together with $|x-a|$:
5 KB (913 words) - 12:24, 12 December 2020
The standard method of regression estimation is based on the use of a polynomial of some degree $ m $,
...ria for testing linearity and for choosing the degree of the approximating polynomial.
12 KB (1,774 words) - 13:04, 13 January 2024
is a polynomial operator, i.e., $ L _ {n} f $
is a polynomial of degree less than or equal to $ n $,
7 KB (1,011 words) - 09:07, 28 June 2022
of such a formula are the zeros of a polynomial $ p _ {n} ( x) $
3 KB (407 words) - 12:57, 19 December 2020
...[[#References|[6]]]). Nevertheless, an algebraic algebra that satisfies a polynomial identity is locally finite. It is not known (1989) whether an algebraic div
3 KB (451 words) - 11:08, 22 July 2016
...roject management and scheduling problems, can be solved efficiently, in a polynomial number of computational steps (see, e.g., [[#References|[a1]]]–[[#Referen
...kerson–Kelley approach as applied to Problem 5 is possible, leading to its polynomial-bounded solution [[#References|[a7]]].
13 KB (2,058 words) - 14:54, 7 June 2020
...ingular projective model of the affine curve $y^2=f(x)$, where $f(x)$ is a polynomial without multiple roots of odd degree $n$ (the case of even degree $2k$ may
2 KB (360 words) - 18:17, 22 November 2014
known sequence. Thus, this process corresponds to polynomial
scheme for constructing the interpolation polynomial.
11 KB (1,808 words) - 21:52, 5 March 2012
...a homogeneous equation $f(X,Y,Z)=0$ in $P^2$ is equal to the degree of the polynomial $f$. See [[Algebraic curve|Algebraic curve]] for the definition of genus, a
3 KB (391 words) - 13:15, 7 April 2023
...Witt theorem|Birkhoff–Witt theorem]], the associated graded algebra is the polynomial algebra.
3 KB (378 words) - 19:39, 5 June 2020
...number of equality- and inequality-type constraints imposed on the Taylor polynomial of the object under investigation.
# All sets $M^k_\pm,M^k_0$ are semialgebraic (defined by finitely many polynomial equalities and inequalities);
7 KB (1,080 words) - 09:14, 12 December 2013
is a homogeneous ideal in the polynomial ring $ k [ X _ {0}, \dots, X _ {n} ] $.
3 KB (364 words) - 08:09, 13 July 2022
...the Artin approximation property if and only if for every finite system of polynomial equations $f$ over $A$ the set of its solutions in $A$ is dense, with respe
...aving a solution in $A ^ { \prime }$ can be enlarged to a finite system of polynomial equations over $A$ having a solution in $A ^ { \prime }$, for which one may
16 KB (2,447 words) - 00:27, 15 February 2024
and the Alexander polynomial is given by
3 KB (418 words) - 07:38, 18 March 2023
...A$ is equivalent to finding the $n$ zeros of its associated characteristic polynomial
...es these eigenvalues, without having to explicitly form the characteristic polynomial $p _ { n } ( z )$ above and then to find its zeros? This was first consider
10 KB (1,639 words) - 07:05, 15 February 2024
can be written as a polynomial in $ X $
3 KB (398 words) - 08:11, 6 June 2020
...respond, respectively, to the nodes of a Chebyshev polynomial or a uniform polynomial. The gas-dynamical functions are piecewise linearly approximated along each
6 KB (913 words) - 16:58, 7 February 2011
Kravchuk's name in its French form was attached to the orthonormal polynomial
normal, the method of moments in statistics, and orthogonal polynomial systems.
10 KB (1,403 words) - 16:21, 10 March 2024
One constructs the interpolation polynomial $ P $
is a polynomial of degree $ \leq n $,
27 KB (3,799 words) - 17:14, 12 November 2023
In terms of polynomial solvability, UET scheduling for uniform machines (processors), where machin
3 KB (392 words) - 17:16, 7 February 2011
...an algebraic group $G$ coincide with the irreducible components. For every polynomial homomorphism $\phi$ of algebraic groups one has $\phi(G^0)=\phi(G)^0$. If $
3 KB (419 words) - 11:16, 18 October 2014
where the polynomial $f(x,y)$ defines a curve of genus $\geq1$ (Siegel in 1929 proved that the n
...rm of degree $\geq3$ and the value of the square-free kernel of an integer polynomial [[#References|[2]]].
6 KB (940 words) - 18:12, 23 November 2014
...nction of $n$ is majorized by a desired complexity bound, like a linear or polynomial function. A more general approach would be to have a single probability dis
...y distribution|Density of a probability distribution]]) can be computed in polynomial time ($\hat { \mu } ( X _ { i } ) = \sum _ { X _ { j } \leq X _ { i } } \mu
21 KB (3,198 words) - 18:47, 11 December 2020
Suppose one is given a polynomial
...-to-one correspondence between polynomials and differential operators with polynomial coefficients and may be extended to significantly broader classes of operat
13 KB (1,836 words) - 14:55, 7 June 2020
...he set of decision problems accepted by Turing machines using at most some polynomial number of steps, $P = \cup _ { k = 1 } ^ { \infty } \operatorname { DTIME }
...f decision problems accepted by a non-deterministic Turing machine in some polynomial time, $\operatorname { NP} = \operatorname { NTIME} [ n ^ { O ( 1 ) } ]$. F
22 KB (3,250 words) - 17:43, 1 July 2020
...ted in the discovery, by P. Shor, of an efficient (that is, consuming only polynomial resources) algorithm for factoring large numbers and for computing discrete
...legacyimages/q/q130/q130020/q13002035.png"/> of languages computable using polynomial working space). Thus, a proof of the important conjecture that $\mathbf{BQP
15 KB (2,154 words) - 17:45, 1 July 2020
be the polynomial algebra over $ F _{0} $
if there is a differential polynomial $ c \in F \{ (y _{j} ) _ {j \in J} \} $
13 KB (1,981 words) - 20:02, 17 December 2019
...ting) local polynomial satisfies (1) at the zeros of a Legendre or Lobatto polynomial (cf. [[Legendre polynomials|Legendre polynomials]]). The numerical properti
...can be found in [[#References|[a14]]]. Collocation methods in certain non-polynomial spline spaces have been studied in [[#References|[a10]]]. A general analysi
12 KB (1,710 words) - 19:39, 18 January 2024
...f the advances provided by the Seiberg–Witten equations concerns Donaldson polynomial invariants for four-dimensional manifolds (see also below).
==Polynomial invariants.==
16 KB (2,663 words) - 10:57, 13 February 2024
in the polynomial should be even. Thus, an expression in Wu generators is obtained for any el
...u generators can be expressed in Pontryagin classes as follows. First, the polynomial is written in elementary symmetric functions of the variables $ x _ {1}
14 KB (1,987 words) - 19:15, 16 January 2024
2) The polynomial $ y = P _ {n} ( x) $
is some polynomial.
8 KB (1,132 words) - 14:19, 17 March 2023
...there is a neighborhood $V$ of $x_0$ and a sequence $P_n$ of homogeneous [[Polynomial|polynomials]] of degree $n$ in $k$ variables such that
*$y^\alpha$ denotes the polynomial $y_1^{\alpha_1} y_2^{\alpha_2} \ldots y_k^{\alpha_k}$.
6 KB (1,048 words) - 21:19, 14 January 2021
...elations which worked successfully in classical knot theory (when defining polynomial invariants of links in $\mathbf{R} ^ { 3 }$, cf. also [[Link|Link]]).
3) The skein module based on the Kauffman polynomial relation is denoted by $S _ { 3 , \infty }$ and is known to be free for $M
11 KB (1,644 words) - 16:45, 1 July 2020
...corresponding Nullstellensatz. This is a theorem describing the form of a polynomial $g$ depending on finitely many other polynomials $f_1,\dots,f_m$, provided
3 KB (454 words) - 20:57, 22 December 2018
is polynomial bounded if and only if there is a polynomial $ P ( n) $
...he collection of problems for which there exists an algorithm operating in polynomial time. One can always identify a [[Decision problem|decision problem]] with
20 KB (3,105 words) - 17:46, 4 June 2020
...d in the real line (cf. also [[Borel measure|Borel measure]]; [[Polynomial|Polynomial]]).
...</td></tr><tr><td valign="top">[a4]</td> <td valign="top"> D.C. Lewis, "Polynomial least square approximations" ''Amer. J. Math.'' , '''69''' (1947) pp. 27
10 KB (1,461 words) - 08:27, 16 March 2024
arbitrary constants and are expressed linearly in terms of a polynomial of degree $ \kappa - 1 $
...ty conditions are satisfied, which can be expressed linearly in terms of a polynomial with variable coefficients.
10 KB (1,509 words) - 08:11, 6 June 2020
...ometric sums: the problem of the distribution of the fractional parts of a polynomial (cf. [[Fractional part of a number|Fractional part of a number]]), and the
H. Weyl studied the distribution of the fractional parts of a polynomial
10 KB (1,485 words) - 14:56, 7 June 2020
...somorphism of the tangent spaces. Locally, an étale morphism is given by a polynomial with non-zero derivative.
3 KB (413 words) - 19:38, 5 June 2020
be a polynomial of degree $ m $.
is non-singular. The Lagrange interpolation polynomial for a function $ f ( x) $
17 KB (2,639 words) - 04:38, 29 December 2021
...he [[Sturm theorem|Sturm theorem]] gives a quantifier-free criterion for a polynomial to admit a root in a given interval, cf. also [[Real closed field|Real clos
3 KB (435 words) - 09:40, 26 November 2016
for certain functions $f$, $\theta$, typically polynomial functions.
3 KB (414 words) - 19:08, 31 July 2014
then the orthogonal polynomial $ P _ {n} ( x) $
3 KB (417 words) - 11:10, 4 January 2021
polynomial with leading coefficient 1 and remaining coefficients from
3 KB (458 words) - 04:43, 4 January 2022
...{\phi_i\}_{i=1}^n$ in a space $C(Q)$ with the property that no non-trivial polynomial in this system has more than $n-1$ distinct zeros. An example of a Chebyshe
3 KB (491 words) - 10:46, 15 August 2014
...re operator on the boundary and $\mathcal{L} ( M , g )$ the Hirzebruch $L$-polynomial associated to the Riemannian metric on $M$.
3 KB (412 words) - 16:58, 1 July 2020
...references to the varied and numerous relations among Hankel matrices and polynomial computations, see [[#References|[a24]]], [[#References|[a9]]], [[#Reference
...n }$ is given by $H _ { n } ^ { - 1 } = B ( q , t )$, where $t ( z )$ is a polynomial of degree less than $n$ that satisfies the Bezout equation $t ( z ) p ( z )
14 KB (2,126 words) - 16:45, 1 July 2020
is equal to the Hermite polynomial $ H _ {n} ( x) $(
3 KB (414 words) - 22:10, 5 June 2020
which means that it is a root of a monic polynomial with coefficients in $ A _ \alpha $,
3 KB (464 words) - 05:59, 19 March 2022
...using a recursive procedure. Often, this recursive procedure uses a Taylor polynomial to replace the function involved in the equation; the most popular elementa
...tive or integral of the function underlying the data is needed. However, a polynomial of degree $n-1$ is generally required to satisfy a set of $n$ conditions, a
29 KB (4,373 words) - 17:21, 2 January 2021
is isomorphic to the polynomial ring $ k [ X _ {1} \dots X _ {n} ] $.
3 KB (469 words) - 08:10, 6 June 2020
of the polynomial $ Q ( z) $
3 KB (479 words) - 17:43, 16 December 2020
is called polynomial. For a polynomial allocation one can also introduce central, right and left domains, and limi
it has a distribution defined by a polynomial allocation, it is possible to use limit theorems for $ \mu _ {0} $
7 KB (1,068 words) - 16:23, 4 March 2022
...ial) gives the corresponding symmetrized tensor (respectively, symmetrized polynomial). Cf. also [[Symmetrization (of tensors)|Symmetrization (of tensors)]].
8 KB (1,112 words) - 08:24, 6 June 2020
...the same expression, i.e., without increasing the degree of the colouring polynomial.
...ricted to given types is also expressed as equality to $0$ of a low-degree polynomial $P(x)$ of a node $x=x_1,\dots,x_k$, its neighbours, and their colours. Over
11 KB (1,701 words) - 10:26, 24 August 2014
...ularity. If, in addition, there are no resonances, then one can choose the polynomial normal form to be linear. If $ \lambda $
7 KB (1,085 words) - 08:13, 6 June 2020
is an equivariant polynomial mapping from $ X $
3 KB (511 words) - 13:16, 7 April 2023
...undamental theorem on the existence of a root of an arbitrary non-constant polynomial over the field of complex numbers.
3 KB (511 words) - 08:04, 6 June 2020
As in the one variable case these provide explicit polynomial approximants for the more-variable Weierstrass approximation and Stone–We
4 KB (588 words) - 10:58, 29 May 2020
and its inverse can be written as a polynomial in $ a $.
3 KB (489 words) - 02:30, 11 July 2022
The ratio between the degree of the polynomial $ P _ {n} (f; x) $
3 KB (474 words) - 10:58, 29 May 2020
...roximation can be increased at the expense of increasing the degree of the polynomial; however, this complicates the approximating process and increases the diff
then a (polynomial) spline of degree $ r $
29 KB (4,328 words) - 18:47, 5 April 2020
is a polynomial in $ x $
is the highest degree of the terms of this polynomial. For instance, the ellipse $ x ^ {2} / a ^ {2} + y ^ {2} / b ^ {2} =
13 KB (2,001 words) - 02:12, 1 March 2022
in the polynomial $ s _ \mu W _ {0} $.
4 KB (519 words) - 16:18, 5 February 2022
''polynomial distribution''
4 KB (522 words) - 13:11, 6 January 2024
A proposition in this language is called a reduced polynomial if it is of the form $ \mathfrak A _{1} + \dots + \mathfrak A _{s} $,
...ynomial. Any formula of the algebra of logic may be converted to a reduced polynomial by identity transformations. The equality $ \mathfrak A = \mathfrak B $
27 KB (3,984 words) - 15:58, 22 December 2019
...-function|$E$-function]]s), satisfying a linear differential equation with polynomial coefficients (cf. [[Siegel method]]). Gel'fond [[#References|[8]]] and T. S
7 KB (1,017 words) - 17:25, 12 November 2023
...lated with the following fixed-point theorem: For every Magari algebra and polynomial $ f ( x ) $
...bra. Moreover, the solution is unique. It is called the fixed point of the polynomial $ f ( x ) $
13 KB (1,762 words) - 11:49, 12 January 2021
...oint property is equivalent to dismantlability and thus can be verified in polynomial time [[#References|[a6]]], [[#References|[a11]]].
9 KB (1,338 words) - 12:00, 9 January 2016
is a polynomial of degree $ m \geq 5 $
3 KB (476 words) - 09:04, 8 October 2023
...atrix shows that the octacode is an extended cyclic code with parity-check polynomial $\widetilde M(x)=x^3+2x^2+x-1$, which reduced modulo $2$ yields $M(x)=x^3+x
3 KB (459 words) - 17:32, 17 April 2016
...racteristic 0 is isomorphic to the cohomology of the corresponding de Rham polynomial complex over $ k $.
4 KB (531 words) - 17:01, 1 July 2020
...er to classify the representation-tame self-injective standard algebras of polynomial growth in [[#References|[a8]]]. In this case tilted algebras of representat
...">[a8]</td> <td valign="top"> A. Skowroński, "Selfinjective algebras of polynomial growth" ''Math. Ann.'' , '''285''' (1989) pp. 177–199</td></tr><tr><td
9 KB (1,240 words) - 08:04, 25 November 2023
...rder. The order of the determinant of the principal symbol (characteristic polynomial) in \eqref{2} is 4, while in \eqref{1} it is 8.
...means of auxiliary functions to an equivalent system with a characteristic polynomial of degree 4 (see [[#References|[7]]]).
8 KB (1,200 words) - 17:04, 14 February 2020
[[Galois group|Galois group]] of a polynomial. If
is a polynomial with coefficients $a_i$ in some field $K$ and $\xi_1,\dots,\xi_n$ are the r
9 KB (1,519 words) - 07:41, 30 November 2014
A homogeneous polynomial of degree 3 in several variables with coefficients in some fixed field or r
3 KB (526 words) - 11:44, 12 October 2023
...acceleration and preconditioning acceleration. Both methods often involve polynomial iterations
is some polynomial chosen from some knowledge or estimation of the eigenvalues of $ A $.
12 KB (1,681 words) - 06:15, 28 March 2023
modulo its radical is a polynomial ring in $ n $
4 KB (568 words) - 14:10, 31 December 2020
...re precisely, for any compact set $K\subset D$ and $\epsilon>0$ there is a polynomial $p(z)$ with complex coefficients such that $|f(z)-p(z)|<\epsilon$ for all $
3 KB (531 words) - 18:58, 1 May 2014