of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more...

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

In mathematics, the degree of a polynomial is the highest of the degrees of the polynomial's monomials (individual terms) with non-zero coefficients. The...

also be non-numerical objects such as polynomials, square matrices, functions, and power series. Formally, a ring is a set endowed with two binary operations...

known as ring theory, a free algebra is the noncommutative analogue of a polynomial ring since its elements may be described as "polynomials" with non-commuting...

ideal of a multivariate polynomial ring, graded by the total degree. The quotient by an ideal of a multivariate polynomial ring, filtered by the total...

or zero of each polynomial in Jα More specifically, Jα is the kernel of the ring homomorphism from F[x] to E which sends polynomials g to their value...

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...

Gröbner basis is a particular kind of generating set of an ideal in a polynomial ring K[x1, ..., xn] over a field K. A Gröbner basis allows many important...

ring to which the coefficients of the polynomial and its possible factors are supposed to belong. For example, the polynomial x2 − 2 is a polynomial with...

group multiplication. the commutative algebra K[x] of all polynomials over K (see polynomial ring). algebras of functions, such as the R-algebra of all real-valued...

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...

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...

Gauss, is a theorem about polynomials over the integers, or, more generally, over a unique factorization domain (that is, a ring that has a unique factorization...

polynomials in X {\displaystyle X} form a ring denoted F [ X , X − 1 ] {\displaystyle \mathbb {F} [X,X^{-1}]} . They differ from ordinary polynomials...

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

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...

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 algebra, a monic polynomial is a non-zero univariate polynomial (that is, a polynomial in a single variable) in which the leading coefficient (the...

commutative algebra. Prominent examples of commutative rings include polynomial rings; rings of algebraic integers, including the ordinary integers Z...

field F is algebraically closed if every non-constant polynomial in F[x] (the univariate polynomial ring with coefficients in F) has a root in F. As an example...

and K-modules are identical. If K is a field, and K[x] a univariate polynomial ring, then a K[x]-module M is a K-module with an additional action of x...

a principal ideal domain such as the integers, or a (multivariate) polynomial ring over a field (this is the Quillen–Suslin theorem). Projective modules...

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

particular the ring of integers, polynomial rings, and rings of algebraic integers in number fields), and many general theorems on rings rely heavily on...

ring of real-valued polynomial functions defined on V can be identified with the quotient ring R[X, Y] / (X2 − Y3), and this is the coordinate ring of V...

r_{2}=0} . For a commutative ring R, the units of the polynomial ring R[x] are the polynomials p ( x ) = a 0 + a 1 x + ⋯ + a n x n {\displaystyle...

resultant of two polynomials is a polynomial expression of their coefficients that is equal to zero if and only if the polynomials have a common root...

mathematics the monomial basis of a polynomial ring is its basis (as a vector space or free module over the field or ring of coefficients) that consists of...

determining whether or not an element in a polynomial ring is irreducible. For example, take an irreducible polynomial f ( x 1 , … , x n ) {\displaystyle f(x_{1}...