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Finiteness criteria for the solutions of the sequence of semi-decomposable form equations and inequalities

Si Duc Quang

TL;DR

This work addresses finiteness criteria for solutions to sequences of semi-decomposable forms over a number field, focusing on $S$-integer solutions and $S$-unit proportionality. It develops a Schmidt-type inequality for families of homogeneous polynomials with distributive constant $\Delta$, bounding sums of Weil functions by $(\Delta(\tfrac{m}{2}+1)^2+\varepsilon)h(x)$ and enabling finiteness results when the semi-decomposable degree $\ell$ satisfies $\ell> d\Delta(\tfrac{m}{2}+1)^2$. The main result proves that, under these conditions and for $\lambda<\ell-d\Delta(\tfrac{m}{2}+1)^2$, there are no infinite sequences of $\mathcal{O}_S^*$-non-proportional $x_n$ with $0<\prod_{v\in S}\|F_n(x_n)\|_v\le c\,H_S^{\lambda}(x_n)$ and $h(F_n)=o(h(x_n))$, and provides a corollary for fixed $F$ with $\deg F> d\Delta(\tfrac{m}{2}+1)^2$ having only finitely many such solutions. These results generalize and unify prior finiteness theorems (e.g., Györy–Ru, Ji–Yan–Yu) via the distributive-constant framework, offering a broad tool for Diophantine finiteness in families of semi-decomposable forms.

Abstract

In this paper, we give some finiteness criteria for the solutions of the sequence of semi-decomposable form equations and inequalities, where the semi-decomposable form is factorized into a family of homogeneous polynomials with the distributive constant exceeding a certain number.

Finiteness criteria for the solutions of the sequence of semi-decomposable form equations and inequalities

TL;DR

This work addresses finiteness criteria for solutions to sequences of semi-decomposable forms over a number field, focusing on -integer solutions and -unit proportionality. It develops a Schmidt-type inequality for families of homogeneous polynomials with distributive constant , bounding sums of Weil functions by and enabling finiteness results when the semi-decomposable degree satisfies . The main result proves that, under these conditions and for , there are no infinite sequences of -non-proportional with and , and provides a corollary for fixed with having only finitely many such solutions. These results generalize and unify prior finiteness theorems (e.g., Györy–Ru, Ji–Yan–Yu) via the distributive-constant framework, offering a broad tool for Diophantine finiteness in families of semi-decomposable forms.

Abstract

In this paper, we give some finiteness criteria for the solutions of the sequence of semi-decomposable form equations and inequalities, where the semi-decomposable form is factorized into a family of homogeneous polynomials with the distributive constant exceeding a certain number.
Paper Structure (3 sections, 5 theorems, 36 equations)

This paper contains 3 sections, 5 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Auxiliary result
  3. Semi-decomposable form equations and inequalities

Key Result

Theorem 1.3

Let $\ell,m,d$ be positive integers. Let $k'$ be a finite extension of $k$ and $S \subset$$M_{k}$ be a finite set containing all archimedean places. For $n=1,2, \ldots$, let $F_{n}({\bf x})=F_{n}(x_{0}, \ldots, x_{m}) \in \mathcal{O}_{S}[{\bf x}]$ be a sequence of semi-decomposable forms of degree $ and

Theorems & Definitions (8)

  • Definition 1.1
  • Theorem 1.3
  • Corollary 1.6
  • Theorem 2.1: see Qpcf
  • Theorem 3.1
  • proof
  • proof : Proof of Theorem \ref{['1.3']}
  • Theorem 3.8