Polynomials assuming only local prime powers

2025/11/11 by Koprowski, Przemysław
#11C08 #FOS: Mathematics #Number Theory (math.NT) #Primary: 11S05 #Secondary: 12J25

paper · doi:10.48550/arxiv.2511.08100

Abstract

We study the class of polynomials that map a local field (i.e., the completion of a number field at a non-Archimedean place) into the subset of its p-th powers, where p is the residue characteristic of the field in question. We present a characterization of such polynomials and show that this class is always much broader than the class of p-th powers of polynomials.

Citations

• p-adic root separation and the discriminant of integer polynomials
• The Ax-Kochen Theorem: an application of model theory to algebra

Related

• Polynomials which are locally reducible
• Iterates of prime producing polynomials and their Galois groups
• Genetics of polynomials over local fields
• Almost universality of a sum of norms
• Polynomials whose nth powers have prescribed multiple-of-nth-degree coefficients