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Prime ideal divisors of parametric recurrence sequences

Darsana N, Sudhansu Sekhar Rout

TL;DR

This work analyzes parametric linear recurrence sequences $U_n(\zeta)$ obtained by specializing at roots of unity and develops effective, explicit arithmetic information. By harnessing dominance analysis of the characteristic roots and powerful Baker-type results on linear forms in logarithms, the authors prove exponential lower bounds for the largest prime ideal divisor and the radical of $(U_n(\zeta))$, and an effective upper bound for the $S$-part of $U_n(\zeta)$. They also derive an effective finiteness result for Diophantine equations of the form $U_n(\zeta)=w_1+\cdots+w_r$ with $S$-unit unknowns, thereby extending the reach of Diophantine techniques to parametric recurrences evaluated at roots of unity. The results are stated under non-degeneracy and non-exceptionality hypotheses on the roots and are accompanied by explicitly computable constants, contributing to the interplay between recurrence sequences, algebraic number theory, and $S$-unit equations.

Abstract

We prove new arithmetic results for parametric linear recurrence sequences specialized at roots of unity, denoted by $(U_n(ζ))_{n\geq 0}$. In particular, we obtain exponential lower bounds for the largest prime ideal divisor and norm of the radical of the principal ideal generated by $U_n(ζ)$. We further derive an effective upper bound for the $S$-part of $U_n(ζ)$, showing that it is strictly smaller than a fixed power of its absolute norm for sufficiently large $n$. Finally, we establish an effective finiteness result for Diophantine equations of the form $U_n(ζ)=w_1+\cdots+w_r,$ with $S$-unit variables.

Prime ideal divisors of parametric recurrence sequences

TL;DR

This work analyzes parametric linear recurrence sequences obtained by specializing at roots of unity and develops effective, explicit arithmetic information. By harnessing dominance analysis of the characteristic roots and powerful Baker-type results on linear forms in logarithms, the authors prove exponential lower bounds for the largest prime ideal divisor and the radical of , and an effective upper bound for the -part of . They also derive an effective finiteness result for Diophantine equations of the form with -unit unknowns, thereby extending the reach of Diophantine techniques to parametric recurrences evaluated at roots of unity. The results are stated under non-degeneracy and non-exceptionality hypotheses on the roots and are accompanied by explicitly computable constants, contributing to the interplay between recurrence sequences, algebraic number theory, and -unit equations.

Abstract

We prove new arithmetic results for parametric linear recurrence sequences specialized at roots of unity, denoted by . In particular, we obtain exponential lower bounds for the largest prime ideal divisor and norm of the radical of the principal ideal generated by . We further derive an effective upper bound for the -part of , showing that it is strictly smaller than a fixed power of its absolute norm for sufficiently large . Finally, we establish an effective finiteness result for Diophantine equations of the form with -unit variables.
Paper Structure (7 sections, 9 theorems, 115 equations)

This paper contains 7 sections, 9 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Notation and Results
  3. Preliminaries
  4. Proof of Theorems
  5. Proof of Theorem \ref{['thm1']}
  6. Proof of Theorem \ref{['thm2']}
  7. Proof of Theorem \ref{['thm3']}

Key Result

Theorem 2.1

Let $(U_n(\zeta))_{n\geq 0}$ be a linear recurrence sequence of the desired structure. Then for all but at most $d^2(2k^3/3 + 22k^2)$ elements $\zeta \in \mathbb{U} \setminus \mathcal{C}_{\alpha,f}$, we have and where $C_1$ and $C_2$ are computable constant depending on $(U_n)_{n\geq 0}, \zeta, D$ and $K$.

Theorems & Definitions (13)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Lemma 3.1: matveev2000explicit
  • Lemma 3.2: matveev2000explicit
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • ...and 3 more