Knot polynomial (15386 bytes)
1: [[Category:Knot theory]]
5: ...t]]. The coefficients are the important part; the polynomial is not meant to be evaluated, but merely a way of...
7: ...var>, these are actually [[Laurent series|Laurent polynomials]] in <var>x^1/n</var> for various <var...
11: ...o communicate than a knot, or even a drawing of a knot.
13: ...d is sufficiently discriminating, two complicated knots can be checked for identity algorithmically.

All one polynomial (1029 bytes)
1: ...ystem|binary]]). The AOP is a 1-[[equally spaced polynomial]].
15: *The AOP is [[irreducible polynomial|irreducible]] if and only if ''m'' + 1 is prime a...
16: *The only AOP that is a [[primitive polynomial]] is ''x''² + x + 1
21: [[Category:Polynomials]]

Polynomial (15087 bytes)
1: ...[mathematics]], '''polynomial functions''', or '''polynomials''', are an important class of simple and [[smoot...
3: ... used extensively in [[numerical analysis]] for [[polynomial interpolation]] or to [[numerical integration|num...
5: ...cs)|matrix]]. In [[graph theory]] the [[chromatic polynomial]] of a [[graph (mathematics)|graph]] encodes the ...
7: ...omials and provide more flexibility than ordinary polynomials when defining simple and smooth functions. They ...
11: ... to the [[complex number]]s, every (non-constant) polynomial has a root: this is the statement of the [[fundam...

Polynomial sequence (657 bytes)
1: ... of the corresponding polynomial. Various special polynomial sequences are known by [[eponym]]s; among these a...
4: * [[Abel polynomials]]
5: * [[Bell polynomials]]
6: * [[Bernoulli polynomials]]
7: * [[Chebyshev polynomials]]

Irreducible polynomial (4657 bytes)
6: of [[polynomial]]s with coefficients in
7: ''F'' is denoted by <math>F[x]</math>. A polynomial
11: polynomials from <math>F[x]</math>.
17: its [[Galois group]], and its irreducible polynomials in depth.
21: It is helpful to compare irreducible polynomials to

Polynomial hierarchy (7794 bytes)
1: In [[computational complexity theory]], the '''polynomial hierarchy''' is a hierarchy of [[complexity class...
5: ... two equivalent definitions of the classes of the polynomial hierarchy: the "oracle" characterization and the ...
7: For the oracle definition of the polynomial hierarchy, define
17: ...sup>NP</sup> is the class of problems solvable in polynomial time with an oracle for some problem in NP.
19: ...h>L</math> be a language, let <math>p</math> be a polynomial, and define

Polynomial time (1597 bytes)
1: ...re the time, ''m''(''n''), is no greater than a [[polynomial function]] of the [[problem size]], ''n''.
5: .... [[Exponential time]] is one example of a super-polynomial time.
7: ...lass of decision problems that can be verified in polynomial time is known as '''[[NP (complexity)|NP]]'''. Eq...
8: polynomial time on a [[non-deterministic Turing machine]] (N...
11: == Subclasses of polynomial time ==

Hurwitz polynomial (495 bytes)
1: ...s, the real part of every zero is negative. These polynomials are named for [[Adolf Hurwitz]].
3: A simple example of a Hurwitz polynomial is the following:
11: [[Category:Polynomials]]

Trigonometric polynomial (2214 bytes)
1: ...re used similar to the [[monomial basis]] for a [[polynomial]].
3: The trigonometric polynomials are used in [[trigonometric interpolation]] to [...
10: ...'' of degree ''N''. Using [[Euler's formula]] the polynomial can be rewritten as
18: is called '''real trigonometric polynomial''' of degree ''N''.
24: ...trigonometric polynomials''. Thus a trigonometric polynomial can be considered a periodic function on the [[re...

Ehrhart polynomial (2877 bytes)
1: ...integral [[polytope]]s have associated '''Ehrhart polynomials''' which encode some geometrical information abo...
3: ...art showed in [[1967]] that ''L'' is a rational [[polynomial]] of degree ''n'' in ''t'', i.e. there exist [[ra...
14: The Ehrhart polynomial of the [[interior (topology)|interior]] of a clos...
20: * Ricardo Diaz, Sinai Robins: ''The Ehrhart polynomial of a lattice ''n''-simplex'', Electronic Research...

Bernstein polynomial (4947 bytes)
1: ...is a [[linear combination]] of '''Bernstein basis polynomials'''.
3: ...al stability|numerically stable]] way to evaluate polynomials in Bernstein form is [[de Casteljau's algorithm]...
5: .... With the advent of computer graphics, Bernstein polynomials, restricted to the interval [0,1], became import...
9: The ''n'' + 1 '''Bernstein basis polynomials''' of degree ''n'' are defined as
13: ...]] for the [[vector space]] <math>\Pi_n</math> of polynomials of degree ''n''.

Polynomial interpolation (13137 bytes)
1: ... (statistics)|sampling]]), and you want to find a polynomial which goes exactly through these points.
5: ...>''i''</sub> are the same we are trying to find a polynomial p of degree ''n'' with the property
8: The '''unisolvence theorem''' states that the polynomial ''p'' of degree ''n'' is uniquely defined by the ...
13: where <math>\Pi_n</math> is the vector space of polynomials with degree ''n''.
15: ==Constructing the interpolation polynomial==

Newton polynomial (3389 bytes)
1: ...ion polynomial''' because the coefficients of the polynomial are calculated using [[divided differences]].
3: ...olynomial. The more precise name is interpolation polynomial in the Newton form.
9: ...form''' is [[linear combination]] of Newton basis polynomials
13: with the Newton basis polynomials defined as
25: Thus the Newton polynomial can be written as

Lagrange polynomial (4081 bytes)
1: ... the [[polynomial interpolation|interpolation]] [[polynomial]] for a given set of data points in the '''Lagran...
3: ...olynomial. The more precise name is interpolation polynomial in the Lagrange form.
6: ...h all 4 control points, and each ''scaled'' basis polynomial passes through its respective control point and i...
10: ...''' is a [[linear combination]] of Lagrange basis polynomials
14: with the Lagrange basis polynomials defined as

Characteristic polynomial (4210 bytes)
1: ...trix]], its '''characteristic polynomial'''. This polynomial encodes several important properties of the matri...
5: Given a square matrix ''A'', we want to find a polynomial whose roots are precisely the eigenvalues of ''A'...
6: ...ntries are ''a'', ''b'', ''c'' the characteristic polynomial will be
12: ...he eigenvalues of ''A''. Since this function is a polynomial in &lambda;, we're done.
16: ..., denoted by ''p''_''A''(''t''), is the polynomial defined by

Minimal polynomial (1197 bytes)
1: ...ree such that ''p''(''A'')=0. Any other non-zero polynomial ''f'' with ''p''(''A'') = 0 is a multiple of ''p'...
5: #&lambda; is a root of the [[characteristic polynomial]] of ''A'',
12: ...polynomial is irreducible, and any other non-zero polynomial ''f'' with ''f''(&alpha;) = 0 is a multiple of ''...
14: [[Category:Polynomials]]

Polynomial ring (4134 bytes)
1: ...ebra]], a '''polynomial ring''' is the [[set]] of polynomials in one or more variables with coefficients in a ...
3: ==Definition of a polynomial==
5: ...bles (see [[polynomial]]), or in other words, a ''polynomial function''.
7: ...[[integer]]s [[modular arithmetic|modulo]] 2, the polynomial
9: ...expect ''P''(''X'') to be different than the zero polynomial.

Separable polynomial (2675 bytes)
1: In [[mathematics]], an [[irreducible polynomial]] P(X) is '''separable''' if its roots in an [[al...
15: ... field ''K''), then ''P''(''x'') is an [[additive polynomial]].

Generic polynomial (2560 bytes)
1: ...eld F, with a <math>\mathbb{Q}</math>-''generic'' polynomial, generic relative to the rational numbers, being ...
3: ...roup. However, not all Galois groups have generic polynomials, a counterexample being the cyclic group of orde...
5: ==Groups with generic polynomials==
9: is a generic polynomial for S_n.
11: ... by eight, and Smith explicitly constructs such a polynomial in case n is not divisible by eight.

Symmetric polynomial (1493 bytes)
1: ...at if some of the variables get interchanged, the polynomial stays the same.
9: ... <math>X_1</math> and <math>X_2</math> we get the polynomial <math>X_2 - X_1</math> which is not the same thin...
11: == The building blocks for symmetric polynomials ==
12: ...les can be obtained from the elementary symmetric polynomials via several multiplications and additions. More ...
13: ...les is a polynomial of the n elementary symmetric polynomials in these variables''.

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