Polynomials Inducing the Zero Function on Local Rings

For a Noetherian local ring pR, mq having a finite residue field of cardinality q, we study the connections between the ideal N pRq of Rrxs, which is the set of polynomials that vanish on R, and the ideal N pmq, the polynomials that vanish on m, using polynomials of the form πpxq ±q i1 px ciq, 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 pRq may be obtained from a generating set for N pmq by composing with πpxq.