The paucity problem for certain symmetric Diophantine equations

TL;DR

This paper investigates the scarcity of non-trivial solutions to certain symmetric Diophantine equations involving elementary symmetric polynomials, under specific degree and sum conditions, highlighting the limitations on solutions in these cases.

Contribution

It establishes new results on the paucity of solutions for a class of symmetric Diophantine systems with particular degree and sum constraints.

Findings

Non-diagonal solutions are scarce under given conditions.

The results apply to systems involving elementary symmetric polynomials.

Provides bounds on the number of solutions for these equations.

Abstract

Let $\varphi_1,\ldots ,\varphi_r\in \mathbb Z[z_1,\ldots z_k]$ be integral linear combinations of elementary symmetric polynomials with $\text{deg}(\varphi_j)=k_j$ $(1\le j\le r)$, where $1\le k_1<k_2<\ldots <k_r=k$. Subject to the condition $k_1+\ldots +k_r\ge \tfrac{1}{2}k(k-1)+2$, we show that there is a paucity of non-diagonal solutions to the Diophantine system $\varphi_j(\mathbf x)=\varphi_j(\mathbf y)$ $(1\le j\le r)$.