On polynomial equations in a multiplicative subgroup of a field of positive characteristic

In the talk I shall consider linear and polynomial equations over fields of positive characteristic. For example over the field of residuals modulo a prime $p$. I shall investigate upper estimates for the number of solutions of these equations such that they belong to a subgroup of the multiplicative group of the field. Then we shall give applications of these estimates to some well-know problem of the additive number theory.