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

TL;DR

This paper extends results on the sum-product problem in finite fields by analyzing polynomial value sets over subgroups, establishing lower bounds and structural properties of such sets.

Contribution

It generalizes previous results by providing bounds on polynomial value sets over subgroups and characterizes the structure of sets generating these subgroups.

Findings

Lower bounds on polynomial value set sizes over subgroups.

Structural result relating subgroup representations to set sizes.

Conditions under which polynomial images cover 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 $G$.