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...

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

Gröbner basis is a particular kind of generating set of an ideal in a polynomial ring K [ x 1 , … , x n ] {\displaystyle K[x_{1},\ldots ,x_{n}]} over a field...

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...

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...

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...

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...

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...

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...

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

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...

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...

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...

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...

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...

theorem for polynomial rings, and for Irvin Cohen (1946), who proved the unmixedness theorem for formal power series rings. All Cohen–Macaulay rings have the...

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...

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...

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...

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

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. In other...

algebra S(V) can be identified, through a canonical isomorphism, to the polynomial ring K[B], where the elements of B are considered as indeterminates. Therefore...

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

zeros over an algebraically closed field k of some family of polynomials in the polynomial ring k [ x 1 , … , x n ] . {\displaystyle k[x_{1},\ldots ,x_{n}]...

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 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...

X1, X2, ..., XN] is the polynomial ring in N + 1 variables Xi. The polynomial ring is therefore the homogeneous coordinate ring of the projective space...

Integer-valued polynomial Algebraic equation Factor theorem Polynomial remainder theorem See also Theory of equations below. Polynomial ring Greatest common...

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

algebraic set is the intersection of the zero sets of several polynomials, in a polynomial ring k [ x 1 , … , x n ] {\displaystyle k\left[x_{1},\ldots ,x_{n}\right]}...