Geodesic
On a Polynomial Version of the Sum-Product Problem for Subgroups
Matematičeskie zametki, Tome 113 (2023) no. 1, pp. 3-10 Cet article a éte moissonné depuis la source Math-Net.Ru

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We generalize two results in the papers [1:x003] and [2:x003] about sums of subsets of $\mathbb{F}_p$ to the more general case in which the sum $x+y$ is replaced by $P(x,y)$, where $P$ is a rather general polynomial. In particular, a lower bound is obtained for the cardinality of the range of $P(x,y)$, where the variables $x$ and $y$ belong to a subgroup $G$ of the multiplicative group of the field $\mathbb{F}_p$. We also prove that if a subgroup $G$ can be represented as the range of a polynomial $P(x,y)$ for $x\in A$ and $y\in B$, then the cardinalities of $A$ and $B$ are close in order to $\sqrt{|G|}$ .
Keywords: subgroup, sum-product problem, sumset problem.
Mots-clés : polynomial 
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S. A. Aleshina; I. V. Vyugin. On a Polynomial Version of the Sum-Product Problem for Subgroups. Matematičeskie zametki, Tome 113 (2023) no. 1, pp. 3-10. http://geodesic.mathdoc.fr/item/MZM_2023_113_1_a0/

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