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...

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...

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

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...

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...

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...

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...

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...

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...

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...

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...

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...

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}]...

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

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...

learning with errors over rings and is simply the larger learning with errors (LWE) problem specialized to polynomial rings over finite fields. Because...

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...

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...

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...

monic polynomials in a univariate polynomial ring over a commutative ring form a monoid under polynomial multiplication. Two monic polynomials are associated...

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

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...

where the zi, j are new unknowns. Linear algebra is effective on the polynomial ring R [ x 1 , … , x n ] {\displaystyle R[x_{1},\ldots ,x_{n}]} if and only...

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...

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...

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]}...

mathematics and computer algebra, factorization of polynomials or polynomial factorization expresses a polynomial with coefficients in a given field or in the...

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