Lacunary Polynomial Compositions

TL;DR

This paper investigates polynomial compositions with a limited number of terms, introducing recursive methods, specific results, and applications to Hilbert sets and integer powers in various scales.

Contribution

It presents a recursive approach to characterize polynomial compositions with few terms and explores their applications in number theory and algebra.

Findings

Recursive characterization of polynomial compositions with fixed terms

Applications to Universal Hilbert Sets and linear recurrence relations

Insights into integer perfect powers with few digits in given scales

Abstract

This work is a study of polynomial compositions having a fixed number of terms. We outline a recursive method to describe these characterizations, give some particular results and discuss the general case. In the final sections, some applications to Universal Hilbert Sets generated by closed forms of linear recurrence relations and to integer perfect powers having few digits in their representation in a given scale $x \ge 2$ are provided.