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March 2015 Polynomial values and generators with missing digits in finite fields
Cécile Dartyge, Christian Mauduit, András Sárközy
Funct. Approx. Comment. Math. 52(1): 65-74 (March 2015). DOI: 10.7169/facm/2015.52.1.5


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.


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Cécile Dartyge. Christian Mauduit. András Sárközy. "Polynomial values and generators with missing digits in finite fields." Funct. Approx. Comment. Math. 52 (1) 65 - 74, March 2015.


Published: March 2015
First available in Project Euclid: 20 March 2015

zbMATH: 1321.11012
MathSciNet: MR3326124
Digital Object Identifier: 10.7169/facm/2015.52.1.5

Primary: 11A63
Secondary: 11L99

Keywords: character sums , digits properties , finite fields , generators , polynomials , primitive roots , squares

Rights: Copyright © 2015 Adam Mickiewicz University


Vol.52 • No. 1 • March 2015
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