Polynomier

Polynomier is a mathematical term used here to denote a polynomial function or expression, defined in one or more variables. It describes a finite linear combination of monomials with coefficients drawn from a field or ring.

In one variable, a polynomier has the form p(x) = a0 + a1 x + … + an x^n, with degree n if an ≠ 0; the zero polynomier has no degree. In several variables, it takes the form p(x1, x2, ..., xk) = sum aα x1^α1 x2^α2 … xk^αk, where α is a multi-index with finite support. The set of polomier in a given coefficient ring F forms the polynomial ring F[x], and in several variables F[x1, ..., xk], both of which are commutative rings with unity.

Operations and properties: polynomier are closed under addition, subtraction, and multiplication; constant polynomier correspond to elements of F. Over a field, nonconstant polynomier may be factored into irreducible polynomier (unique up to units and order) in the appropriate polynomial ring, a statement refined by the fundamental theorems of algebra in suitable contexts. Real polynomier can be analyzed by real roots and, in the complex setting, by complex roots which occur in conjugate pairs.

Applications: polynomier are central in algebra and analysis, used in interpolation, approximation, numerical methods, algebraic geometry, and computer algebra systems. They serve as foundational objects for understanding functions, graphs, and their qualitative behavior, such as monotonicity and curvature, via derivatives and factorial or Taylor expansions.

History and terminology: the term polynomier may appear as an archaic or regional variant of polynomial in some texts, or as a didactic designation to emphasize the polynomial structure. See also polynomial, polynomial ring, algebra.