Asymptotics for Pillai's problem with polynomials

TL;DR

This paper derives asymptotic estimates for the number of polynomial power sum pairs with degrees bounded by a growing parameter, extending understanding of polynomial sum behaviors in asymptotic regimes.

Contribution

It provides the first asymptotic analysis for the count of polynomial power sum pairs with degree constraints, generalizing classical results to polynomial settings.

Findings

Asymptotic formula for the number of polynomial power sum pairs

Extension of classical Pillai's problem to polynomial sums

Quantitative bounds on degrees of polynomial sums

Abstract

Let $ a_1(x)p_1(x)^n + \cdots + a_k(x)p_k(x)^n $ as well as $ b_1(x)q_1(x)^m + \cdots + b_l(x) q_l(x)^m $ be two polynomial power sums where the complex polynomials $ p_i(x) $ and $ q_j(x) $ are all non-constant. Then in the present paper we will give an asymptotic for the number of pairs $ (n,m) \in \mathbb{N}^2 $ such that the degree of the sum of these two power sums is between $ 0 $ and $ d $ when $ d $ goes to infinity.