Two Properties of Pseudo-Polynomials over a Galois Field

Let F be the completion, with respect to the degree valuation, of the field of rational functions F-q(x) over F-q, the Galois (finite) field of q elements. A function f : F -> F is integer-valued if f(F-q[x]) subset of F-q[x]. An integer-valued function f is called a pseudo-polynomial if f(M + K) = f(M) (mod K) for all M is an element of F-q[x] and K is an element of F-q[x] \ {0}. Based on an interpolation series introduced by Carlitzin 1935, explicit shapes of pseudo-polynomials are established. Using an asymptotic characterization of polynomials, it is also proved that the set of all pseudo-polynomials is an integral domain but not a unique factorization domain.