Polynomials Inducing the Zero Function on Local Rings

TL;DR

This paper investigates polynomials that vanish on local rings and their ideals, establishing connections between these polynomials, the ring's structure, and properties like Henselianity, nilpotency, and completeness.

Contribution

It introduces pi-polynomials and explores their role in describing zero-vanishing ideals, providing new characterizations and decompositions for these ideals in various local ring contexts.

Findings

Z(R) equals the intersection of ideals generated by pi-polynomials

When R is Henselian, p(R) equals m and Z(R) relates to Z(m) via composition

Explicit descriptions of Z(m) depending on nilpotency index e

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.