Strong paucity in systems of diagonal equations

Abstract: Let $k$ be a natural number with $k\ge 2$, and let $\varepsilon>0$. We consider the number $V_k^*(P)$ of integral solutions of the system of simultaneous Diophantine equations $$x_1^{2j-1}+\ldots +x_{k+1}^{2j-1}=y_1^{2j-1}+\ldots +y_{k+1}^{2j-1}\quad (1\le j\le k).$$ with $1\le x_i,y_i\le P$ $(1\le i\le k+1)$. Writing $L_k^*(P)$ for the number of diagonal solutions with $\{x_1,\ldots ,x_{k+1}\}=\{y_1,\ldots ,y_{k+1}\}$, so that $L_k^*(P)\sim (k+1)!P^{k+1}$, we prove that $$V_k^*(P)-L_k^*(P)\ll P^{\sqrt{8k+9}-1+\varepsilon}.$$ This establishes a strong paucity result improving on earlier work of Brüdern and Robert. Time permitting, we describe analogous results for related problems.

number theory

Audience: researchers in the topic

Combinatorial and additive number theory seminar (CANT 2026)