In ring theory, a branch of mathematics, a ring R is a polynomial identity ring if there is, for some N > 0, an element P ≠ 0 of the free algebra, Z⟨X1...

Polynomial identity may refer to: Algebraic identities of polynomials (see Factorization) Polynomial identity ring Polynomial identity testing This disambiguation...

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...

may also be non-numerical objects such as polynomials, square matrices, functions, and power series. A ring may be defined as a set that is endowed with...

a right primitive ring, then RU is ring isomorphic to R. In (Rowen 1980, p.154), a version is given for polynomial identity rings. The notation Z(R)...

the ring of symmetric functions is a specific limit of the rings of symmetric polynomials in n indeterminates, as n goes to infinity. This ring serves...

Bézout's identity (also called Bézout's lemma), named after Étienne Bézout who proved it for polynomials, is the following theorem: Bézout's identity—Let a...

mathematics, Newton's identities, also known as the Girard–Newton formulae, give relations between two types of symmetric polynomials, namely between power...

mathematics, the ring of polynomial functions on a vector space V over a field k gives a coordinate-free analog of a polynomial ring. It is denoted by...

approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts...

the "identity" of the ring, and the phrases "ring with unity" or a "ring with identity" may be used to emphasize that one is considering a ring instead...

abbreviated as GCD) of two polynomials is a polynomial, of the highest possible degree, that is a factor of both the two original polynomials. This concept is analogous...

In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues...

between two matrix polynomials, which holds for the specific matrices in question. A matrix polynomial identity is a matrix polynomial equation which holds...

i for the variable X in the polynomial p) is a surjective ring homomorphism. The kernel of f consists of all polynomials in R[X] that are divisible by...

elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed...

coordinates over any basis. A polynomial of degree 0 is always homogeneous; it is simply an element of the field or ring of the coefficients, usually called...

Ore extensions, with R any commutative polynomial ring, σ the identity ring endomorphism, and δ the polynomial derivative. Ore algebras are a class of...

content by a unit of the ring of the coefficients (and the multiplication of the primitive part by the inverse of the unit). A polynomial is primitive if its...

quotient ring Z / n Z {\displaystyle \mathbb {Z} /n\mathbb {Z} } (which has n {\displaystyle n} elements). Now consider the ring of polynomials in the variable...

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...

for any 2-by-2-matrices x and y. Generic matrix ring Formanek, Edward (1991). The polynomial identities and invariants of n×n matrices. Regional Conference...

R_{i}=0} for i ≠ 0. This is called the trivial gradation on R. The polynomial ring R = k [ t 1 , … , t n ] {\displaystyle R=k[t_{1},\ldots ,t_{n}]} is...

for its applications, such as homological properties and polynomial identities. Commutative rings are much better understood than noncommutative ones. Algebraic...

The Chebyshev polynomials are two sequences of orthogonal polynomials related to the cosine and sine functions, notated as T n ( x ) {\displaystyle T_{n}(x)}...

commutative ring satisfies a certain identity of degree 2n. It was proved by Amitsur and Levitsky (1950). In particular matrix rings are polynomial identity rings...

addition and multiplication. Ring theory is the study of rings, exploring concepts such as subrings, quotient rings, polynomial rings, and ideals as well as...

polynomials as a sum of squares of rational functions Metric tensor Polynomial ring Polynomial SOS, the representation of a non-negative polynomial as...

commutative ring. The rational, real and complex numbers form fields. If R {\displaystyle R} is a given commutative ring, then the set of all polynomials in the...

solutions, similarly as polynomial algebras are used for the study of algebraic varieties, which are solution sets of systems of polynomial equations. Weyl algebras...