Polynomials Inducing the Zero Function on Local Rings

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 $\pi(x) = \prod_{i = 1}^{q} (x - c_{i})$, where $c_{1}, \ldots, c_{q}$ 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 $\pi(x)$.