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arXiv · 2608.26349

On polynomial expanders with many variables

M. Z. Garaev·S. V. Konyagin
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Abstract

For a fixed integer $n\ge 2,$ we consider the homogeneous polynomial $$ P(x_1, x_2, \ldots, x_{n+2})=\sum_{i=1}^{n} (x_2-x_1)^{i-1} x_1^{n-i} x_{i+2}. $$ We prove that, for any finite set $A$ of complex numbers, $$ \Bigl|\bigl\{P(x_1,x_2,\ldots,x_{n+2}): \, x_i\in A\bigr\}\Bigr|\gg |A|^{n}. $$ The implicit constant in $\gg$ may depend only on $n.$

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M. Z. Garaev, S. V. Konyagin. 2026-08-26. On polynomial expanders with many variables. https://arxiv.org/abs/2608.26349

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