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arXiv · 2107.12238

Paucity problems and some relatives of Vinogradov's mean value theorem

Trevor D. Wooley
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Abstract

When $k\ge 4$ and $0\le d\le (k-2)/4$, we consider the system of Diophantine equations \[ x_1^j+\ldots +x_k^j=y_1^j+\ldots +y_k^j\quad (1\le j\le k,\, j\ne k-d). \] We show that in this cousin of a Vinogradov system, there is a paucity of non-diagonal positive integral solutions. Our quantitative estimates are particularly sharp when $d=o(k^{1/4})$.

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Trevor D. Wooley. 2021-07-26. Paucity problems and some relatives of Vinogradov's mean value theorem. https://doi.org/10.1017/s0305004123000166

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