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## Solutions of Polynomial Equations in Subgroups of F_p^*

Arnold Mathematical Journal. 2019. Vol. 5. No. 1. P. 105-121.
Макарычев С. В., Vyugin I. V.

We present an upper bound on the number of solutions of an algebraic equation P(x,y)=0 where x and y belong to the union of cosets of some subgroup of the multiplicative group κ∗ of some field of positive characteristic. This bound generalizes the bound of Corvaja and Zannier (J Eur Math Soc 15(5):1927–1942, 2013) to the case of union of cosets. We also obtain the upper bounds on the generalization of additive energy.