On a Polynomial Version of the Sum-Product Problem for Subgroups

We generalize two results in the papers [1:x003] and [2:x003] 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 $\sqrt{|G|}$ .