On some polynomial version on the sum-product problem for subgroups

Abstract

We generalize two results about subgroups of multiplicative group of finite field of prime order. In particular, the lower bound on the cardinality of the set of values of polynomial $P(x,y)$ is obtained under the certain conditions, if variables $x$ and $y$ belong to a subgroup $G$ of the multiplicative group of the filed of residues. Also the paper contains a proof of the result that states that if a subgroup $G$ can be presented as a set of values of the polynomial $P(x,y)$, where $x\in A$, and $y\in B$ then the cardinalities of sets $A$ and $B$ are close (in order) to a square root of the cardinality of subgroup $G$.