Solutions of Polynomial Equations in Subgroups of $$\\mathbb {F}_p^*$$Fp∗

Abstract

We present an upper bound on the number of solutions of an algebraic equation $$P(x,y)=0$$P(x,y)=0 where x and y belong to the union of cosets of some subgroup of the multiplicative group $$\\kappa ^*$$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.