polynomiet

Polynomiet is a term that appears in some mathematical texts to denote a polynomial-like algebraic object. In most contexts, the standard term polynomial is preferred. When used, polynomiet usually refers to an element of a polynomial ring over a commutative ring, i.e., a finite formal sum of monomials with coefficients from that ring.

Formally, a polynomiet in n variables over a commutative ring R is an expression of the form sum over multi-indices α of c_α X^α, where c_α ∈ R and X^α denotes X_1^{α_1}...X_n^{α_n}. Only finitely many coefficients c_α are nonzero. The collection of all such polynomiets with fixed R and variables X_1, ..., X_n forms the polynomial ring R[X_1, ..., X_n].

Operations on polynomiet are defined as in standard polynomial rings: addition and multiplication are performed by combining like terms and applying the usual distributive law. The set of polynomiets with a given R and fixed variables is a ring, and if R is a field, it is an integral domain. Key notions include the degree (the maximum total degree among nonzero terms), the leading coefficient, and evaluation (substituting elements of R for the variables).

Polynomiet can be univariate or multivariate. Constant polynomiets correspond to elements of the base ring R. They can be manipulated by standard algorithms for polynomials, such as division, factorization, and GCD computations, provided the coefficient ring allows it.

Applications include solving equations, polynomial interpolation, algebraic geometry, and coding theory. See also: polynomial, polynomial ring, multivariate polynomial, interpolation, factorization.