polynomiales

Polynomiales is a term used in some mathematical contexts to denote polynomials, the standard objects studied in algebra and algebraic geometry. A polynomiale in one variable x with coefficients in a ring R is a finite sum a0 + a1 x + a2 x^2 + ... + an x^n, where each coefficient ai lies in R and an ≠ 0. The set of all polynomiales in one variable over R is denoted R[x], and in several variables x1, ..., xk it is denoted R[x1, ..., xk]. Polynomiales form a ring under the usual addition and multiplication of polynomials.

Key properties include the notion of degree, defined as the largest exponent with a nonzero coefficient (for the zero polynomiale the degree is often left undefined or set to −∞). If R is a field, then R[x] is a principal ideal domain and a unique factorization domain; every nonzero polynomiale factors uniquely into irreducible polynomials up to units. When R is not a field, many of these statements are modified, but R[x] still inherits a rich algebraic structure.

Roots of polynomiales are central to many theories. Over a field, a nonconstant polynomiale has roots in an algebraic closure, as guaranteed by the fundamental theorem of algebra in the univariate case. Multivariate polynomiales define algebraic varieties via their common zero sets.

Applications of polynomiales span many areas, including interpolation and approximation (such as fitting data with polynomials), coding theory, numerical analysis, and algebraic geometry, where the solution sets of systems of polynomials describe geometric objects. The study of polynomiales encompasses arithmetic, factorization, and computational algorithms for division, greatest common divisors, and factorization.