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Zeros and $S$-units in sums of terms of recurrence sequences in function fields

Darsana N, S. S. Rout

TL;DR

This work studies effective finiteness questions for Diophantine equations involving sums of linear recurrence sequences over function fields and $S$-units. By combining the Brownawell–Masser height bound with a careful analysis of minimal vanishing subsums, the authors prove that sums of $r\ge 2$ terms of a simple, non-degenerate recurrence must have bounded indices when equal to an $S$-unit, and they classify the finite versus infinite set of solutions to $U_n+V_m+W_\ell=0$ for three such sequences, identifying precise algebraic obstructions that yield infinite families. The results yield effective constants bounding the indices and extend the understanding of polynomial–exponential Diophantine equations in function-field settings, complementing existing number-field and positive-characteristic outcomes. Overall, the paper advances the theory of $S$-units in linear recurrence frameworks over function fields and provides a computable, structural account of when and how infinite solution families arise.

Abstract

Let $(U_n)_{n\geq 0}$ be a non-degenerate linear recurrence sequence with order at least two defined over a function field and $\mathcal{O}_S^*$ be the set of $S$-units. In this paper, we use a result of Brownawell and Masser to prove effective results related to the Diophantine equations concerning linear recurrence sequences and $S$-units. In particular, we provide a finiteness result for the solutions of the Diophantine equation $U_{n_1} + \cdots + U_{n_r} \in \mathcal{O}_S^*$ in nonnegative integers $n_1, \ldots, n_r$. Furthermore, we study the finiteness result of the Diophantine equation $U_n+V_m+W_\ell = 0$ in $(n, m, \ell)\in \N^3$, where $U_n,V_m,W_\ell$ are simple linear recurrence sequences in the function field.

Zeros and $S$-units in sums of terms of recurrence sequences in function fields

TL;DR

This work studies effective finiteness questions for Diophantine equations involving sums of linear recurrence sequences over function fields and -units. By combining the Brownawell–Masser height bound with a careful analysis of minimal vanishing subsums, the authors prove that sums of terms of a simple, non-degenerate recurrence must have bounded indices when equal to an -unit, and they classify the finite versus infinite set of solutions to for three such sequences, identifying precise algebraic obstructions that yield infinite families. The results yield effective constants bounding the indices and extend the understanding of polynomial–exponential Diophantine equations in function-field settings, complementing existing number-field and positive-characteristic outcomes. Overall, the paper advances the theory of -units in linear recurrence frameworks over function fields and provides a computable, structural account of when and how infinite solution families arise.

Abstract

Let be a non-degenerate linear recurrence sequence with order at least two defined over a function field and be the set of -units. In this paper, we use a result of Brownawell and Masser to prove effective results related to the Diophantine equations concerning linear recurrence sequences and -units. In particular, we provide a finiteness result for the solutions of the Diophantine equation in nonnegative integers . Furthermore, we study the finiteness result of the Diophantine equation in , where are simple linear recurrence sequences in the function field.
Paper Structure (6 sections, 8 theorems, 88 equations)

This paper contains 6 sections, 8 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Notation and results
  3. Preliminaries
  4. Proof of Theorem \ref{['thm1']}
  5. Proof of Theorem \ref{['thm2']}
  6. Proof of Theorem \ref{['thm2']}:

Key Result

Theorem 2.1

Let $L$ be a function field in one variable over an algebraically closed field $k$ of zero characteristic and $S$ be a finite set of valuations on $L$. Let $(U_n)_{n\geq 0}$ be a simple linear recurrence sequence of order at least two defined over $L$. Assume that $(U_n)_{n\geq 0}$ is non-degenerate for $n_1>\cdots> n_r$ implies $\max(n_1, \ldots, n_r)\leq C$.

Theorems & Definitions (15)

  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.3
  • Lemma 3.1
  • proof
  • Proposition 3.2: Brownawell-Masser
  • Lemma 3.3
  • Lemma 5.1
  • proof
  • Proposition 5.2
  • ...and 5 more