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Characterization of norm and quasi-norm forms in S-adic setting

George Tomanov

TL;DR

This work analyzes norm and quasi-norm forms over a number field $F$ in the $S$-adic setting by examining their values on $S$-integers. It develops an $S$-adic dynamical framework, including a reduction to the $m=n$ case and an $S$-adic Mahler criterion, to characterize when an $S$-form is determined by its discrete image on $\mathcal{O}_S^n$. The main result shows that an $S$-form is an $S$-norm form or an $S$-quasi-norm form if and only if $f(\mathcal{O}_S^n)$ is discrete and $f$ does not represent $0$ over $F$ non-trivially, equivalently iff the orbit $H\pi(e)$ is compact in the $S$-adic homogeneous space $G/\Gamma$. This extends prior results for $F=\mathbb{Q}$ and provides a broader dynamical characterization of norm vs. quasi-norm forms in the $S$-adic setting, with a constructive description of quasi-norm forms via totally real field data.

Abstract

The goal of the present paper is to characterize the norm and quasi-norm forms defined over an arbitrary number field F in terms of their values at the S-integer points, where S is a finite set of valuations of F containing the archimedean ones. In this way we generalize the main result of the recent paper [T5], where the notion of a quasi-norm form is introduced when F = Q and S is a singleton. In complement, we exhibit some relations with problems and results in this area of research.

Characterization of norm and quasi-norm forms in S-adic setting

TL;DR

This work analyzes norm and quasi-norm forms over a number field in the -adic setting by examining their values on -integers. It develops an -adic dynamical framework, including a reduction to the case and an -adic Mahler criterion, to characterize when an -form is determined by its discrete image on . The main result shows that an -form is an -norm form or an -quasi-norm form if and only if is discrete and does not represent over non-trivially, equivalently iff the orbit is compact in the -adic homogeneous space . This extends prior results for and provides a broader dynamical characterization of norm vs. quasi-norm forms in the -adic setting, with a constructive description of quasi-norm forms via totally real field data.

Abstract

The goal of the present paper is to characterize the norm and quasi-norm forms defined over an arbitrary number field F in terms of their values at the S-integer points, where S is a finite set of valuations of F containing the archimedean ones. In this way we generalize the main result of the recent paper [T5], where the notion of a quasi-norm form is introduced when F = Q and S is a singleton. In complement, we exhibit some relations with problems and results in this area of research.

Paper Structure

This paper contains 11 sections, 13 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Normalized valuations and norms.
  4. $S$-adic compactness criterium.
  5. Algebraic groups and tori.
  6. A reduction result
  7. On the stabilizer of a polynomial.
  8. Proof of Theorem \ref{['reduction thm']}.
  9. Proof of Theorem \ref{['thm++']}
  10. A compactness criterium.
  11. Completion of the proof of Theorem \ref{['thm++']}.

Key Result

Theorem 1.1

Toma2 Let $f(\vec{x})$ be a decomposable over $F_S$ form such that $f({\mathcal{O}}_S^m)$ is a discrete subset of $F_S$. Then $f(\vec{x}) = ag(\vec{x})$, where $g(\vec{x}) \in F[\vec{x}]$ and $a \in F_S^*$.

Theorems & Definitions (13)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Lemma 2.1
  • Theorem 2.2
  • Proposition 2.3
  • Proposition 3.1
  • Proposition 4.1
  • ...and 3 more