polinomot

Polinomot is a mathematical construct that generalizes the concept of a polynomial by allowing the coefficients of a polynomial to be polynomials themselves. Formally, if R is a commutative ring, a polinomot in the indeterminate t with coefficients in R[X] is an expression of the form P(t) = sum_{i=0}^n f_i(X) t^i, where each f_i(X) belongs to R[X]. Equivalently, polinomots are elements of the polynomial ring R[X][t], which is isomorphic to the multivariate polynomial ring R[X,t].

Operations on polinomots are defined by the usual polynomial rules, with t treated as the primary variable and the f_i(X) as coefficients. Addition and multiplication are performed coefficientwise, in the ring R[X]. The degree of a polinomot is the largest exponent of t with a nonzero coefficient, and a total-degree notion can combine the degrees in X with the t-degree by considering deg_X f_i and deg_t.

Examples help illustrate the idea. In Z[X], the expression P(t) = (X^2 + 1) + (3X) t + (X) t^2 is a polinomot. More generally, any two-variable polynomial f(X,t) = a_0(X) + a_1(X) t + a_2(X) t^2 with a_i(X) ∈ Z[X] can be viewed as a polinomot, where the coefficients of t are polynomials in X.

Applications of polinomots appear in modeling parametric families of polynomials, symbolic computation, and aspects of algebraic geometry where one studies how polynomial behavior changes with a parameter. The term polinomot is not standardized in literature and is used here as a descriptive label for a polynomial whose coefficients are themselves polynomials. See also polynomials, polynomial rings, multivariate polynomials, and symbolic computation.