Polynomials Inducing the Zero Function on Local Rings

Mark W. Rogers
Cameron Wickham

Abstract

For a Noetherian local ring $(R, \f{m})$ having a finite residue field of cardinality q, we study the connections between the ideal \zf{R} of R[x], which is the set of polynomials that vanish on R, and the ideal \zf{\f{m}}, the polynomials that vanish on \f{m}, using polynomials of the form π (x)=∏qi=1(x-ci), where c1,…,cq is a set of representatives of the residue classes of \f{m}. In particular, when R is Henselian we show that a generating set for \zf{R} may be obtained from a generating set for \zf{\f{m}} by composing with π (x).