Polynomials Inducing the Zero Function on Local Rings

Abstract

For a Noetherian local ring (R, m) having a finite residue field of cardinality q, we study the connections between the ideal Z(R) of R[x], which is the set of polynomials that vanish on R, and the ideal Z(m), the polynomials that vanish on m, using what we call pi-polynomials: polynomials of the form p(x) = \prod_{i = 1}^{q} (x - c_i), where c1, ..., cq is a set of representatives of the residue classes of m. When R is Henselian we prove that p(R) = m and show that a generating set for Z(R) may be obtained from a generating set for Z(m) by composing with p(x). When m is principal and has index of nilpotency e, we prove that if e \leq q then Z(m) = (x, m)^e, and if e = q + 1 then Z(m) = (x, m)^e + (x^q - m^{q - 1} x). When R is finite, we prove that Z(R) = \cap_{i = 1}^{q} Z(c_i + m) is a minimal primary decomposition. We determine when Z(R) is nonzero, regular, or principal, respectively, and do the same for Z(m). We prove that when R is complete, repeated application of p(x) + x to elements of R will produce a sequence converging to the roots of p(x). We show that Z(R) is the intersection of the principal ideals generated by the pi-polynomials.