A Solution to the Polynomial Hensel Code Conversion Problem

The polynomial Hensel code of a rational function a(x)/ b(x) ϵ F(x), F is a field, is the pair (c(x) d-1(x) mod Xr, n); r is a positive integer and a(x)/ b(x) = (c(x))xn such that c(x) and d(x) have nonzero constant terms. Such a representation scheme was proposed, in analogy with the Hensel code representations of rational numbers, to facilitate arithmetic operations on rational functions and control intermediate expressions well. The difficulty with this scheme was the conversion of such a code to rational function form. In this correspondence, we have given sufficient conditions under which this can be done and have described an algorithm for effecting the conversion. We have also discussed an application, namely, the reduction of a rational function to its simplest form.