On squares in special sets of finite fields

Abstract

We consider the linear vector space formed by the elements of the finite fields ${\mathbb{F}}_q$ with $q=p^r$ over ${\mathbb{F}}_p$. Let $\{a_1,\dots ,a_r\}$ be a basis of this space. Then the elements $x$ of ${\mathbb{F}}_q$ have a unique representation in the form ${\sum }_{j=1}^r{c}_j{a}_j$ with $c_j\in {\mathbb{F}}_p$. Let $D_1,\dots ,D_r$ be subsets of ${\mathbb{F}}_p$. We consider the set $W=W(D_1,\dots ,D_r)$ of elements of ${\mathbb{F}}_q$ such that $c_j\in D_j$ for all $j=1,\dots ,r$. We give an estimate for the number of squares in the set $W$ which implies an asymptotic formula for this quantity in the case when the sets $D_1,\dots ,D_r$ are "large on average" and a sufficient condition for the existence of squares in the set $W$.

arXiv comment9 pages, in Russian