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Diophantine Equations for Polynomial Recursive Sequences

Darsana N, Sudhansu Sekhar Rout

TL;DR

The paper investigates Diophantine equations of the form $U_n(x)=V_m(y)$ for polynomial power-sum sequences over number fields, leveraging the Bilu–Tichy finiteness criterion to characterize infinite, denominator-bounded families via a composition relation $U_n(x)=V_m(y)=\phi(Q(y))$. It then extends decomposition results to polynomial sequences from third- and second-order recurrences, proving uniform bounds on the degree of the outer polynomial in decompositions $W_n(x)=g(h(x))$ or $u_n(x)+u_m(x)=g(h(x))$ under indecomposability and nontriviality conditions, with proofs relying on heights, $S$-unit bounds, and genus inequalities. The results yield structural constraints on when such polynomial recurrence terms can be decomposed or arise from compositional identities, and they include corollaries that describe the form of infinite integer solutions. Overall, the work extends Diophantine-analytic techniques to higher-order polynomial recurrences and to sums of recurrence terms, providing explicit degree bounds and decomposition criteria with potential for effective computation.

Abstract

We study the Diophantine equation of type $U_n(x)=V_m(y)$, where $(U_n)_{n\geq 0}$ and $(V_m)_{m\geq 0}$ are polynomial power sums defined over a number field $K$. By applying the finiteness criterion of Bilu and Tichy, we show under appropriate assumptions that equation $U_n(x)=V_m(y)$ has infinitely many solutions with bounded $\mathcal{O}_S$-denominator. We also study decomposable polynomials in third and second order linear recurrence sequences. In particular, we show that if $W_n(x)=g(h(x))$ for a simple third order linear recurrence sequence $(W_n(x))_{n\geq 0}$ of complex polynomials, then deg $g$ is bounded. Furthermore, we show that if $(u_{n_1}+u_{n_2})(x)=g(h(x))$ for a binary recurrence sequence $(u_n(x))_{n\geq 0}$ then deg $g$ is bounded.

Diophantine Equations for Polynomial Recursive Sequences

TL;DR

The paper investigates Diophantine equations of the form for polynomial power-sum sequences over number fields, leveraging the Bilu–Tichy finiteness criterion to characterize infinite, denominator-bounded families via a composition relation . It then extends decomposition results to polynomial sequences from third- and second-order recurrences, proving uniform bounds on the degree of the outer polynomial in decompositions or under indecomposability and nontriviality conditions, with proofs relying on heights, -unit bounds, and genus inequalities. The results yield structural constraints on when such polynomial recurrence terms can be decomposed or arise from compositional identities, and they include corollaries that describe the form of infinite integer solutions. Overall, the work extends Diophantine-analytic techniques to higher-order polynomial recurrences and to sums of recurrence terms, providing explicit degree bounds and decomposition criteria with potential for effective computation.

Abstract

We study the Diophantine equation of type , where and are polynomial power sums defined over a number field . By applying the finiteness criterion of Bilu and Tichy, we show under appropriate assumptions that equation has infinitely many solutions with bounded -denominator. We also study decomposable polynomials in third and second order linear recurrence sequences. In particular, we show that if for a simple third order linear recurrence sequence of complex polynomials, then deg is bounded. Furthermore, we show that if for a binary recurrence sequence then deg is bounded.
Paper Structure (9 sections, 12 theorems, 143 equations, 2 tables)

This paper contains 9 sections, 12 theorems, 143 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Notation and results
  3. Auxiliary results
  4. Proof of main results
  5. Proof of Theorem \ref{['thm1']}
  6. Proof of Theorem \ref{['2thm']}
  7. Proof of Theorem \ref{['thm2']}
  8. Proof of Corollary \ref{['corolary1']}:
  9. Concluding Remark

Key Result

Theorem 2.2

Let $K$ be the number field, $S$ be a finite set of places of $K$ containing all Archimedean places, $(U_n(x))_{n\geq 0}$ and $(V_m(y))_{m\geq 0}$ be linear recurrence sequences of the desired structure with power sum representation $U_n(x)=a_1(x)\alpha_1(x)^n+\cdots+a_d(x)\alpha_d(x)^n$ and $V_m(y) has infinitely many solutions $(x,y)\in K\times K$ with a bounded $\mathcal{O}_S$-denominator if an

Theorems & Definitions (21)

  • Remark 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Remark 2.5
  • Remark 2.6
  • Corollary 2.7
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • ...and 11 more