For a Noetherian local ring (R, m) having a finite residue field of cardinality q, we study the connections between the ideal N (R) of R[x], which is the set of polynomials that vanish on R, and the ideal N(m), the polynomials that vanish on m, using polynomials of the form (formula presented) where c1,…, cq is a set of representatives of the residue classes of m. In particular, when R is Henselian we show that a generating set for N (R) may be obtained from a generating set for N (m) by composing with π(x)
