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$#C+1 = 7 : ~/encyclopedia/old_files/data/T094/T.0904230 Trigonometric polynomial,
is called the order of the trigonometric polynomial (provided $ | a _ {n} | + | b _ {n} | > 0 $).
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...$\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]]
37 bytes (3 words) - 18:52, 24 March 2012
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 $n=2$
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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
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
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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
1 KB (187 words) - 21:08, 23 November 2023
The normalized version of the Alexander polynomial (cf. also [[Alexander invariants]]). It satisfies the Conway skein relation
...[Knot theory]]). For $z=\sqrt t-1/\sqrt t$ one gets the original Alexander polynomial (defined only up to $\pm t^i$).
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...blishes relations between the roots and the coefficients of a [[Polynomial|polynomial]]. Let $ f( x) $
be a polynomial of degree $ n $
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''additive polynomial''
A [[polynomial]] over a [[field]] of [[Characteristic of a field|characteristic]] $p \ne 0
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...polynomials $\sigma_k(x_1,\ldots,x_n)$ for $k=0,\ldots,n$ where the $k$-th polynomial is obtained by summing all distinct [[monomial]]s which are products of $k$
...] $S_n$, so that any symmetric polynomial in the $x_i$ can be written as a polynomial in the $\sigma_k$.
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...]]], [[#References|[a2]]] and generalized by L.H. Kauffman (the [[Kauffman polynomial]]; cf. also [[Link]]).
...> <TD valign="top"> R.D. Brandt, W.B.R. Lickorish, K.C. Millett, "A polynomial invariant for unoriented knots and links" ''Invent. Math.'' , '''84''' (1
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...s knot]]) are Neuwirth knots. So is every alternating knot whose Alexander polynomial has leading coefficient $\pm1$.
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The normalized version of the Alexander polynomial (cf. also [[Alexander invariants]]). It satisfies the Conway skein relation
...[Knot theory]]). For $z=\sqrt t-1/\sqrt t$ one gets the original Alexander polynomial (defined only up to $\pm t^i$).
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 $X \subset \mat
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$#C+1 = 32 : ~/encyclopedia/old_files/data/C021/C.0201720 Characteristic polynomial
The polynomial
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...r or polynomial or element of a ring; see [[Divisor (of an integer or of a polynomial)]];
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#REDIRECT [[Divisor (of an integer or of a polynomial)]]
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...f a function $f$ in the form of a [[Trigonometric polynomial|trigonometric polynomial]]
...unction at $2n+1$ preassigned points $x_k$ in the interval $[0,2\pi)$. The polynomial has the form
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For references, see [[Kauffman polynomial|Kauffman polynomial]].
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...nt: 1) $A$ is semi-simple; 2) the [[Minimal polynomial of a matrix|minimal polynomial]] of $A$ has no multiple factors in $F[X]$; and 3) the algebra $F[A]$ is a
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A necessary and sufficient condition for all the roots of a polynomial
...art criterion]], and methods for determining the number of real roots of a polynomial are also known.
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...e $\pm1$. '''Littlewood's problem''' asks how large the values of such a polynomial must be on the [[unit circle]] in the [[complex plane]]. The answer to th
A polynomial
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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$#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 $).
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Every non-trivial knot has a non-trivial [[Jones polynomial]].
...es not hold, as M.B. Thistlethwaite found a $15$-crossing link whose Jones polynomial coincides with a trivial link of two components, cf. Fig.a1. This and simil
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A divisor of a polynomial $A(x)$ is a polynomial $B(x)$ that divides $A(x)$ without remainder (cf. [[Division|Division]]).
...e of $a$, then $a$ is ''irreducible''. For polynomials, see [[Irreducible polynomial]]; for integers, the traditional terminology is [[prime number]].
1 KB (209 words) - 08:06, 26 November 2023