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Polynomials assuming only local prime powers

Przemysław Koprowski

TL;DR

We classify polynomials with coefficients in rings of integers that send the local field $K_\\frakp$ into its $p$-th powers, defining the monoid $\\mathscr{C}(K_\\frakp)$. A concrete characterization links $F$ and its reciprocal $\\mathrm{rev} F$, yielding $F \\in \\mathscr{C}(K) \\iff F, \\mathrm{rev} F \\in \\mathscr{C}(\\mathbb{Z}_K)$, and we show $\\mathscr{C}(K) = \\mathscr{C}(K_\\frakp)$ while $\\mathscr{C}(\\mathbb{Z}_K) = \\mathscr{C}(\\mathcal{O}_\\frakp)$; notably, $\\mathscr{C}(K)$ is strictly contained in $\\mathscr{C}(\\mathbb{Z}_K)$. Decidability is achieved via an explicit algorithm rooted in Ax–Kochen: after a square-free decomposition of $F$, one tests a suitably reduced $F_*$ and its reciprocal, with a finite test guaranteeing membership. Importantly, the class of polynomials mapping onto $K_\\frakp^{\\times p}$ is strictly larger than the submonoid of actual $p$-th powers of polynomials, demonstrated by a constructed $F$ that is not a $p$-th power yet maps all inputs to $p$-th powers; nonetheless, polynomials in $\\mathscr{C}(\\mathbb{Z}_K)$ can be approximated by $p$-th powers on the valuation ring. These results illuminate how local $p$-power behavior interacts with global polynomial structure and provide explicit decision procedures for related classes.

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.

Polynomials assuming only local prime powers

TL;DR

We classify polynomials with coefficients in rings of integers that send the local field into its -th powers, defining the monoid . A concrete characterization links and its reciprocal , yielding , and we show while ; notably, is strictly contained in . Decidability is achieved via an explicit algorithm rooted in Ax–Kochen: after a square-free decomposition of , one tests a suitably reduced and its reciprocal, with a finite test guaranteeing membership. Importantly, the class of polynomials mapping onto is strictly larger than the submonoid of actual -th powers of polynomials, demonstrated by a constructed that is not a -th power yet maps all inputs to -th powers; nonetheless, polynomials in can be approximated by -th powers on the valuation ring. These results illuminate how local -power behavior interacts with global polynomial structure and provide explicit decision procedures for related classes.

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 -th powers, where 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 -th powers of polynomials.

Paper Structure

This paper contains 5 sections, 16 theorems, 56 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminaries
  4. Decidability of the class $\mathscr{C}(K_\mathfrak{p})$
  5. Relations to the $p$-th powers

Key Result

Proposition 3.1

For every $k> ep/(p-1)$ one has $1 + \mathfrak{p}^k\subseteq K_\mathfrak{p}^{\times p}$.

Figures (1)

  • Figure 1: The field $\mathbb{Q}_3$ as a union of classes of third powers.

Theorems & Definitions (33)

  • Proposition 3.1: FV2002
  • Corollary 3.2
  • Proposition 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • Corollary 3.6
  • Proposition 3.7
  • ...and 23 more