On 12/4/09, Eugene Salamin <gene_salamin@yahoo.com> wrote:
Polynomials wear two hats. On the one hand, they are elements of a certain ring R[x] generated by an indeterminate x and a coefficient ring R, and on the other hand they are functions from R to R. When R is a finite field of order q, a polynomial p as a function uniquely determines a polynomial p as ring element only mod the ideal (x^q-x). Furthermore, according to the Lagrange interpolation theorem, every function F_q --> F_q is the same as some polynomial function of degree less than q. So perhaps this notion of combinomials is useful only over infinite rings.
-- Gene
More accurately, when the domain is distinct from the range --- in the application I used this for (sequences over a prime field) , the domain is \N and the range \F_p. WFL