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THE PAUCITY PROBLEM FOR CERTAIN SYMMETRIC DIOPHANTINE EQUATIONS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 108 / Issue 1 / August 2023
- Published online by Cambridge University Press:
- 23 September 2022, pp. 29-39
- Print publication:
- August 2023
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- Article
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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 
$1\le k_1<k_2<\cdots <k_r=k$. Subject to the condition 
$k_1+\cdots +k_r\ge \tfrac {1}{2}k(k-~1)+2$, we show that there is a paucity of nondiagonal solutions to the Diophantine system 
$\varphi _j({\mathbf x})=\varphi _j({\mathbf y})\ (1\le j\le r)$.
