Geodesic
On a~Polynomial Version of the Sum-Product Problem for Subgroups
Matematičeskie zametki, Tome 113 (2023) no. 1, pp. 3-10.

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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/

[1] I. E. Shparlinski, “Additive Decompositions of Subgroups of Finite Fields”, SIAM J. Discrete Math., 27:4 (2013), 1870–1879 | DOI | MR

[2] I. V. Vyugin, I. D. Shkredov, “Ob additivnykh sdvigakh multiplikativnykh podgrupp”, Matem. sb., 203:6 (2012), 81–100 | DOI | MR | Zbl

[3] A. Garcia, J. Voloch, “Fermat curves over nite elds”, J. Number Theory, 30:3 (1988), 345–356 | DOI | MR

[4] D. Heath-Brown, S. Konyagin, “New bounds for Gauss sums derived from k-thpowers, and for Heilbronn's exponential sum”, Q. J. Math., 51:2 (2000), 221–235 | DOI | MR

[5] B. Murphy, M. Rudnev, I. Shkredov, Y. Shteinikov, J. Théor. Nombres Bordeaux, 31:3 (2019), 573–602 | MR

[6] S. Makarychev, I. Vyugin, “Solutions of polynomial equation over $\mathbb{F}_p$ and new bounds of additive energy”, Arnold Math J., 5:1 (2019), 105–121 | DOI | MR

[7] I. V. Vyugin, “Otsenka chisla proobrazov polinomialnogo otobrazheniya”, Matem. zametki, 106:2 (2019), 212–221 | DOI | MR