...$x,y,\dots,w$ 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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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 $
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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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#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 $n=2$
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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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...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 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
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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
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
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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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#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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''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 [[Hilbert polynomial]]
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#REDIRECT [[Linearised 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 [[Elementary symmetric polynomial]]
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#REDIRECT [[Alexander-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)
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
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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
1 KB (159 words) - 21:20, 7 May 2016