...$x,y,\dots,w$ 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.
9 KB (1,497 words) - 10:44, 27 June 2015
..., $P_M(n) = h_M(n) = \dim_K M_n$. 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
2 KB (361 words) - 05:51, 17 April 2024
...[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
...)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 [[Jones-Conway polynomial]]
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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 polynomial
The values of the Taylor polynomial and of its derivatives up to order $n$ inclusive at the point $x=x_0$ coinc
1 KB (233 words) - 17:01, 16 March 2013
#REDIRECT [[Alexander-Conway polynomial]]
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-
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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
$#C+1 = 32 : ~/encyclopedia/old_files/data/C021/C.0201720 Characteristic polynomial
The polynomial
3 KB (434 words) - 11:43, 24 December 2020
#REDIRECT [[Brandt-Lickorish-Millett-Ho polynomial]]
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#REDIRECT [[Jones-Conway polynomial]]
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-
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$#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 $).
1 KB (158 words) - 14:03, 28 June 2020
...$\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
#REDIRECT [[Alexander-Conway polynomial]]
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#REDIRECT [[Jones-Conway polynomial]]
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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
$#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
$#C+1 = 13 : ~/encyclopedia/old_files/data/L060/L.0600810 Lommel polynomial
The polynomial $ R _ {m, \nu } ( z) $
1 KB (213 words) - 04:11, 6 June 2020
#REDIRECT [[Fejér polynomial]]
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...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
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
1 KB (159 words) - 21:20, 7 May 2016
...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 $
5 KB (651 words) - 08:36, 1 July 2022
A trigonometric polynomial of the form
or a similar polynomial in sines. Fejér polynomials are used in constructing continuous functions
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#REDIRECT [[Szegö polynomial]]
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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''
5 KB (801 words) - 20:34, 13 September 2016
''polynomial deviating least from zero''
An algebraic polynomial of degree $n$, with leading coefficient 1, having minimal norm in the space
3 KB (552 words) - 15:05, 14 February 2020
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
7 KB (1,046 words) - 17:02, 1 July 2020
...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
3 KB (577 words) - 13:35, 20 March 2023
...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
1 KB (187 words) - 21:08, 23 November 2023
...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
3 KB (478 words) - 15:26, 30 December 2018
#REDIRECT [[Linearised polynomial]]
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-
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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)
2 KB (276 words) - 00:18, 25 November 2018
''additive polynomial''
A [[polynomial]] over a [[field]] of [[Characteristic of a field|characteristic]] $p \ne 0
303 bytes (51 words) - 19:48, 1 January 2015
...nd and older results. The paper [[#References|[a24]]] is an early study on polynomial convexity.
Polynomial convexity arises naturally in the context of function algebras (cf. also [[
20 KB (3,071 words) - 17:45, 1 July 2020
-
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It is a Laurent polynomial of one variable associated to ambient isotopy classes of unoriented framed
...^ { ( 1 ) } \rangle = - A ^ { 3 } \langle L \rangle$. The Kauffman bracket polynomial is also considered as an invariant of regular isotopy (Reidemeister moves:
7 KB (1,054 words) - 07:42, 10 February 2024
$#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
5 KB (695 words) - 22:14, 5 June 2020
...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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-
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$#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
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 $\pm t^i$).
1 KB (155 words) - 07:11, 24 March 2024
''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
$#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) $
6 KB (907 words) - 08:06, 6 June 2020
...teger programming problems in a fixed number of variables can be solved in polynomial time.
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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
4 KB (571 words) - 16:10, 1 April 2020
''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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...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]]].
6 KB (1,005 words) - 20:14, 15 March 2023
...]]], [[#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
1 KB (162 words) - 08:42, 26 March 2023
...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
1 KB (237 words) - 14:06, 20 March 2023
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]].
1 KB (209 words) - 08:06, 26 November 2023
$#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
''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
1 KB (237 words) - 14:06, 20 March 2023
...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
1 KB (233 words) - 17:01, 16 March 2013
...)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
2 KB (271 words) - 19:29, 2 November 2014
#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 [[Jones-Conway polynomial]]
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#REDIRECT [[Elementary symmetric polynomial]]
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#REDIRECT [[Jones-Conway polynomial]]
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...t–Lickorish–Millett–Ho polynomial]] and the [[Kauffman polynomial|Kauffman polynomial]]:
1 KB (190 words) - 10:58, 26 March 2023
...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)
2 KB (276 words) - 00:18, 25 November 2018
A trigonometric polynomial of the form
or a similar polynomial in sines. Fejér polynomials are used in constructing continuous functions
491 bytes (73 words) - 15:11, 23 April 2014
...[[characteristic polynomial]] and [[Minimal polynomial of a matrix|minimal polynomial]] coincide (up to a factor $\pm1$). Equivalently, for each of its distinct
936 bytes (133 words) - 22:28, 22 November 2016
...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 $\alpha$.
1 KB (180 words) - 16:57, 25 November 2023
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
1 KB (159 words) - 21:20, 7 May 2016
...blishes relations between the roots and the coefficients of a [[Polynomial|polynomial]]. Let $ f( x) $
be a polynomial of degree $ n $
3 KB (355 words) - 14:04, 20 March 2023
''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 $\sigma_k$.
1,001 bytes (159 words) - 20:35, 13 September 2016
...]]], [[#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
1 KB (162 words) - 08:42, 26 March 2023
...s knot]]) are Neuwirth knots. So is every alternating knot whose Alexander polynomial has leading coefficient $\pm1$.
1 KB (166 words) - 20:34, 11 April 2014
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 $\pm t^i$).
1 KB (155 words) - 07:11, 24 March 2024
..., $P_M(n) = h_M(n) = \dim_K M_n$. 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
2 KB (361 words) - 05:51, 17 April 2024
$#C+1 = 32 : ~/encyclopedia/old_files/data/C021/C.0201720 Characteristic polynomial
The polynomial
3 KB (434 words) - 11:43, 24 December 2020
...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
2 KB (269 words) - 15:19, 14 February 2020
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
1 KB (227 words) - 18:07, 12 November 2017
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.
2 KB (262 words) - 12:37, 14 February 2020
...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
1 KB (187 words) - 21:08, 23 November 2023
...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
3 KB (478 words) - 15:26, 30 December 2018
$#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
5 KB (695 words) - 22:14, 5 June 2020
$#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 $).
1 KB (158 words) - 14:03, 28 June 2020
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
3 KB (378 words) - 16:33, 29 March 2024
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]].
1 KB (209 words) - 08:06, 26 November 2023
''polynomial regression''
exist). The regression is called parabolic (polynomial) if the components of the vector $ {\mathsf E} \{ Y \mid X \} = f( x)
2 KB (301 words) - 08:05, 6 June 2020
...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
2 KB (283 words) - 16:38, 30 December 2018
...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
13 KB (1,768 words) - 17:09, 7 February 2011
...le of topology has ultimately been reduced to the single assumption that a polynomial of odd degree with real coefficients has a real root.
2 KB (348 words) - 05:59, 20 August 2014
''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 $
2 KB (328 words) - 19:37, 5 June 2020
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
1 KB (201 words) - 21:31, 5 March 2012
...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]]].
6 KB (1,005 words) - 20:14, 15 March 2023
...perfect matching exists. (It should be noted that this is not the [[Tutte polynomial]] of $G$.)
1 KB (226 words) - 07:28, 14 November 2023
* A [[divisor (of an integer or of a polynomial)]]
148 bytes (22 words) - 17:20, 16 September 2016
...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
3 KB (382 words) - 06:29, 30 May 2020
$#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
4 KB (571 words) - 16:10, 1 April 2020
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
4 KB (619 words) - 08:04, 6 June 2020
...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 $
5 KB (651 words) - 08:36, 1 July 2022
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 $
3 KB (469 words) - 10:15, 29 May 2020
''polynomial deviating least from zero''
An algebraic polynomial of degree $n$, with leading coefficient 1, having minimal norm in the space
3 KB (552 words) - 15:05, 14 February 2020
...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
1 KB (248 words) - 08:08, 17 March 2023
...l and its generalizations (e.g. the [[Jones–Conway polynomial|Jones–Conway polynomial]]).
2 KB (295 words) - 08:04, 19 March 2023
...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.
2 KB (373 words) - 09:18, 12 December 2013
''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
3 KB (435 words) - 17:32, 5 June 2020
...is unipotent if and only if its [[Characteristic polynomial|characteristic polynomial]] is $ ( x - 1) ^ {n} $.
1 KB (163 words) - 08:27, 6 June 2020
...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
3 KB (496 words) - 07:37, 18 March 2023
...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
2 KB (268 words) - 15:02, 14 February 2020
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
1 KB (152 words) - 19:37, 17 November 2023
...pt of a limit transition), and makes sense for any coefficient ring. For a polynomial
of a polynomial, then $ x _ {0} $
2 KB (246 words) - 19:39, 5 June 2020
...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 $1 , \dots , f$
4 KB (549 words) - 15:30, 1 July 2020
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.
4 KB (628 words) - 21:20, 11 December 2017
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
3 KB (579 words) - 17:05, 7 November 2016
...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.
1 KB (173 words) - 07:29, 14 November 2023
...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
3 KB (574 words) - 18:14, 14 June 2023
A method for simultaneously calculating all roots of a polynomial. Suppose that the roots $ r _ {1} \dots r _ {n} $
of the polynomial
5 KB (723 words) - 07:55, 14 January 2024
It is a Laurent polynomial of one variable associated to ambient isotopy classes of unoriented framed
...^ { ( 1 ) } \rangle = - A ^ { 3 } \langle L \rangle$. The Kauffman bracket polynomial is also considered as an invariant of regular isotopy (Reidemeister moves:
7 KB (1,054 words) - 07:42, 10 February 2024
...$ 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
2 KB (380 words) - 19:41, 15 November 2023
...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
3 KB (488 words) - 14:57, 14 February 2020
...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.
1 KB (191 words) - 18:12, 7 February 2017
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) $
4 KB (533 words) - 10:58, 29 May 2020
...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
4 KB (579 words) - 16:57, 1 July 2020
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)
3 KB (520 words) - 17:30, 16 December 2020
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
7 KB (1,046 words) - 17:02, 1 July 2020
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''
5 KB (801 words) - 20:34, 13 September 2016
...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
2 KB (331 words) - 11:31, 17 September 2017
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:
3 KB (536 words) - 09:04, 27 December 2013
...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
6 KB (1,000 words) - 17:05, 7 July 2014
...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 $
3 KB (397 words) - 20:25, 10 January 2021
...$x,y,\dots,w$ 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.
9 KB (1,497 words) - 10:44, 27 June 2015
$#C+1 = 13 : ~/encyclopedia/old_files/data/L060/L.0600810 Lommel polynomial
The polynomial $ R _ {m, \nu } ( z) $
1 KB (213 words) - 04:11, 6 June 2020
...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
3 KB (577 words) - 13:35, 20 March 2023
...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
2 KB (348 words) - 18:14, 12 November 2017
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:
1 KB (229 words) - 15:40, 14 February 2020
$#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) $
6 KB (907 words) - 08:06, 6 June 2020
''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
6 KB (1,011 words) - 06:01, 30 September 2023
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
2 KB (236 words) - 13:17, 14 February 2020
...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
3 KB (484 words) - 05:48, 17 April 2024
...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
4 KB (698 words) - 15:52, 10 April 2023
...$ 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
6 KB (1,030 words) - 17:29, 12 November 2023
for every [[Trigonometric polynomial|trigonometric polynomial]] $ Q $
for every (algebraic) [[Polynomial|polynomial]] $ Q $
5 KB (674 words) - 07:59, 6 June 2020
...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
6 KB (866 words) - 18:56, 24 January 2024
...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
2 KB (269 words) - 16:56, 1 July 2020
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 $,
4 KB (573 words) - 18:05, 26 January 2022
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.
5 KB (859 words) - 11:30, 12 May 2022
...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 $
6 KB (972 words) - 21:59, 29 January 2020
...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
2 KB (416 words) - 12:41, 19 August 2014
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
692 bytes (106 words) - 17:21, 28 October 2014
such that for any polynomial
If for any such polynomial $ \Phi ( P) > 0 $,
2 KB (245 words) - 08:07, 6 June 2020
...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
2 KB (305 words) - 16:55, 16 October 2014
...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 ) $
3 KB (414 words) - 08:29, 6 June 2020
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]]).
7 KB (974 words) - 10:10, 11 November 2023
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 $x^n-2$.
10 KB (1,645 words) - 17:08, 14 February 2020
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}
5 KB (768 words) - 22:11, 5 June 2020
...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
3 KB (372 words) - 22:15, 5 June 2020
...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
4 KB (680 words) - 13:40, 30 December 2018
[[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
535 bytes (87 words) - 09:09, 7 December 2012
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.
2 KB (376 words) - 08:01, 6 June 2020
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 $.
5 KB (778 words) - 02:00, 4 January 2022
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
4 KB (661 words) - 16:10, 1 April 2020
...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
4 KB (576 words) - 10:55, 2 July 2020
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]];
760 bytes (111 words) - 19:55, 21 December 2015
...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
667 bytes (110 words) - 08:23, 28 April 2023
...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
2 KB (366 words) - 06:13, 10 April 2023
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 $
6 KB (850 words) - 22:11, 5 June 2020
...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
4 KB (749 words) - 18:32, 2 November 2014
...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
5 KB (851 words) - 14:57, 30 July 2014
...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 $,
6 KB (845 words) - 01:33, 5 March 2022
[[Kharitonov polynomial theory]] |
1 KB (118 words) - 17:47, 25 April 2012
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
8 KB (1,192 words) - 08:49, 10 November 2023
...] (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
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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 $
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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
4 KB (708 words) - 13:51, 20 March 2023
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
7 KB (1,101 words) - 14:50, 27 January 2024
...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]]).
3 KB (487 words) - 18:21, 11 April 2017
...^ { 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:
7 KB (968 words) - 16:56, 1 July 2020
...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
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...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
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...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
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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 $
3 KB (503 words) - 20:41, 21 December 2019
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} $
10 KB (1,519 words) - 02:51, 21 March 2022
...; \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
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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
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is the Laguerre polynomial (cf. [[Laguerre polynomials]]) of degree $ n $.
is the generalized Laguerre polynomial (see [[#References|[4]]]).
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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) $
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All roots of a polynomial
with a constant matrix $A$, the characteristic polynomial of which is $P ( z )$ (see [[#References|[4]]]).
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...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'
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$#C+1 = 103 : ~/encyclopedia/old_files/data/B110/B.1100250 Bell polynomial
is a homogeneous polynomial of degree $ k $
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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} $
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is a polynomial of degree $(m-1)$ with respect to $n$.
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