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Value Sets of Sparse Polynomials

Part of: Arithmetic problems. Diophantine geometry Finite fields and commutative rings (number-theoretic aspects)

Published online by Cambridge University Press:  24 September 2019

Igor E. Shparlinski  and
José Felipe Voloch
Igor E. Shparlinski
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia Email: igor.shparlinski@unsw.edu.au
José Felipe Voloch
Affiliation:
School of Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand Email: felipe.voloch@canterbury.ac.nz
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Abstract

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We obtain a new lower bound on the size of the value set $\mathscr{V}(f)=f(\mathbb{F}_{p})$ of a sparse polynomial $f\in \mathbb{F}_{p}[X]$ over a finite field of $p$ elements when $p$ is prime. This bound is uniform with respect to the degree and depends on some natural arithmetic properties of the degrees of the monomial terms of $f$ and the number of these terms. Our result is stronger than those that can be extracted from the bounds on multiplicities of individual values in $\mathscr{V}(f)$.

Keywords

sparse polynomialvalue setrational point on curve

MSC classification

Primary: 11T06: Polynomials
Secondary: 14G15: Finite ground fields

Information

Type
Article
Information
Canadian Mathematical Bulletin , Volume 63 , Issue 1 , March 2020 , pp. 187 - 196
Copyright
© Canadian Mathematical Society 2019

Footnotes

Author I. E. S. was supported by ARC Grants DP170100786 and DP180100201.

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