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arXiv · 2211.10500

The paucity problem for certain symmetric Diophantine equations

Trevor D. Wooley
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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 $\varphi_j(\mathbf x)=\varphi_j(\mathbf y)$ $(1\le j\le r)$.

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Trevor D. Wooley. 2022-11-18. The paucity problem for certain symmetric Diophantine equations. https://doi.org/10.1017/s000497272200096x

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