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

We generalize two results in the papers [1] and [2] about sums of subsets of to the more general case in which the sum is replaced by , where is a rather general polynomial. In particular, a lower bound is obtained for the cardinality of the range of , where the variables and belong to a subgroup of the multiplicative group of the field . We also prove that if a subgroup can be represented as the range of a polynomial for and , then the cardinalities of and are close in order to . [ABSTRACT FROM AUTHOR]