Sum of terms of recurrence sequences and $S$-units in the solution sets of norm form equations

TL;DR

This paper establishes finiteness results for solutions of norm form equations involving sums of recurrence sequence terms and $S$-units, using advanced number theory techniques.

Contribution

It provides new finiteness theorems for solutions of norm form equations related to recurrence sequences and $S$-units, expanding understanding of their solution sets.

Findings

Finiteness of sums of recurrence sequence terms in solutions

Finiteness of solutions as sums of fixed $S$-units

Application of polynomial-exponential and $S$-unit equations theory

Abstract

In this paper, we prove two results related to the solutions of norm form equations. Firstly, we give a finiteness result for sums of terms of linear recurrence sequences appearing in the coordinates of solutions of norm form equations. Next, we give a finiteness result concerning solutions of norm form equations representable as sums of $S$-units with a fixed number of terms. To prove these results, we use a deep results concerning the finiteness of the solutions of polynomial-exponential equations and $S$-unit equations.