...$x,y,\dots,w$ 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
...[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
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#REDIRECT [[Alexander-Conway polynomial]]
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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
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#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) $
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#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''
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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
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
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#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)
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''additive polynomial''
A [[polynomial]] over a [[field]] of [[Characteristic of a field|characteristic]] $p \ne 0
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...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 [[
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-
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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 $X \subset \mat
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-
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$#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
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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...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
...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
''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
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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 [[Jones-Conway polynomial]]
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#REDIRECT [[Jones-Conway polynomial]]
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#REDIRECT [[Alexander-Conway polynomial]]
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#REDIRECT [[Elementary symmetric 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 $\sigma_k$.
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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
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.
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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 $
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)]]
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