Polynomial values and generators with missing digits in finite fields

Abstract

We consider the linear vector space formed by the elements of the finite field $\mathbb{F}_q$ with $q=p^r$ over $\mathbb{F}_p$. Then the elements $x$ of $\mathbb{F}_q$ have a unique representation in the form $x=\sum_{j=1}^r c_ja_j$ with $c_j\in\mathbb{F}_p$; the coefficients $c_j$ will be called digits. Let $\mathbb{D}$ be a subset of $\mathbb{F}_p$ with $2\le |\mathbb{D}|<p$. We consider elements $x$ of $\mathbb{F}_q$ such that for their every digit $c_j$ we have $c_j\in\mathbb{D}$; then we say that the elements of $\mathbb{F}_p\setminus\mathbb{D}$ are ``missing digits''. We will show that if $\mathbb{D}$ is a large enough subset of $\mathbb{F}_p$, then there are squares with missing digits in $\mathbb{F}_q$; if the degree of the polynomial $f(x)\in\mathbb{F}_q[X]$ is at least $2$ then it assumes values with missing digits; there are generators $g$ in $\mathbb{F}_q$ such that $f(g)$ is of missing digits.