Abstract
We generalize two results in the papers [1] and [2] 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 \(\sqrt{|G|}\).
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Acknowledgments
In conclusion, the authors express their gratitude to Andrey Volgin and the referee for their useful comments.
Funding
This work was supported by the Russian Science Foundation under grant 19-11-00001.
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Translated from Matematicheskie Zametki, 2023, Vol. 113, pp. 3–10 https://doi.org/10.4213/mzm13530.
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Aleshina, S.A., V’yugin, I.V. On a Polynomial Version of the Sum-Product Problem for Subgroups. Math Notes 113, 3–9 (2023). https://doi.org/10.1134/S0001434623010017
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DOI: https://doi.org/10.1134/S0001434623010017