In algebra, an alternating polynomial is a polynomial f ( x 1 , … , x n ) {\displaystyle f(x_{1},\dots ,x_{n})} such that if one switches any two of the...

being alternating is equivalent to being symmetric). Among all alternating polynomials, the Vandermonde polynomial is the lowest degree monic polynomial. Conversely...

property of this invariant states that the Jones polynomial of an alternating link is an alternating polynomial. This property was proved by Morwen Thistlethwaite...

symmetric polynomial is a polynomial P(X1, X2, ..., Xn) in n variables, such that if any of the variables are interchanged, one obtains the same polynomial. Formally...

_{n}}\end{matrix}}\right]} are alternating polynomials by properties of the determinant. A polynomial is alternating if it changes sign under any transposition...

In computational complexity theory, the polynomial hierarchy (sometimes called the polynomial-time hierarchy) is a hierarchy of complexity classes that...

Newton polynomial, named after its inventor Isaac Newton, is an interpolation polynomial for a given set of data points. The Newton polynomial is sometimes...

symmetric functions are polynomial functions, which are given by the symmetric polynomials. A related notion is alternating polynomials, which change sign...

especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally...

of a Taylor series is a polynomial of degree n that is called the nth Taylor polynomial of the function. Taylor polynomials are approximations of a function...

An alternating Turing machine (or to be more precise, the definition of acceptance for such a machine) alternates between these modes. An alternating Turing...

In mathematics, a generic polynomial refers usually to a polynomial whose coefficients are indeterminates. For example, if a, b, and c are indeterminates...

logarithmic memory by alternating Turing machines. P is also known to be no larger than PSPACE, the class of problems decidable in polynomial space. Again, whether...

theory gives a set of inequalities for alternating sums of Betti numbers in terms of a corresponding alternating sum of the number of critical points N...

In mathematics, a polynomial P(X) over a given field K is separable if its roots are distinct in an algebraic closure of K, that is, the number of distinct...

an alternating group is the group of even permutations of a finite set. The alternating group on a set of n elements is called the alternating group...

simple, noncyclic, normal subgroup, namely the alternating group An. Van der Waerden cites the polynomial f(x) = x5 − x − 1. By the rational root theorem...

concept of alternating planar algebras first appeared in the work of Hernando Burgos-Soto on the Jones polynomial of alternating tangles. Alternating planar...

the mathematical field of knot theory, a knot polynomial is a knot invariant in the form of a polynomial whose coefficients encode some of the properties...

In mathematics, Jacobi polynomials (occasionally called hypergeometric polynomials) P n ( α , β ) ( x ) {\displaystyle P_{n}^{(\alpha ,\beta )}(x)} are...

polynomials with symmetric Galois groups. For n > 4, the symmetric group S n {\displaystyle {\mathcal {S}}_{n}} of degree n has only the alternating group...

In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander...

orthogonal polynomials are the most widely used orthogonal polynomials: the Hermite polynomials, Laguerre polynomials, Jacobi polynomials (including as...

theory, the HOMFLY polynomial or HOMFLYPT polynomial, sometimes called the generalized Jones polynomial, is a 2-variable knot polynomial, i.e. a knot invariant...

hierarchy, and consists of all families of circuits of depth O(1) and polynomial size, with unlimited-fanin AND gates and OR gates (we allow NOT gates...

{\displaystyle \sum _{k=1}^{n-1}k^{p}=1^{p}+2^{p}+3^{p}+\cdots +(n-1)^{p}} as a polynomial in n. In modern notation, Faulhaber's formula is ∑ k = 1 n − 1 k p = 1...

P is a polynomial of degree N having the property described, that is, it gives rise to an error function that has N + 2 extrema, of alternating signs and...

characterization of PSPACE is the set of problems decidable by an alternating Turing machine in polynomial time, sometimes called APTIME or just AP. A logical characterization...

by a polynomial of degree k {\textstyle k} , called the k {\textstyle k} -th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the...

APSPACE, the set of all problems that can be solved by an alternating Turing machine in polynomial space. EXPTIME relates to the other basic time and space...