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Values of polynomials over finite fields

Published online by Cambridge University Press:  17 April 2009

Joachim von zur Gathen
Joachim von zur Gathen
Affiliation:
Department of Computer Science, University of Toronto, Toronto, Ontario M5S 1A4, Canada
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Abstract

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Let q be a prime power, Fq a field with q elements, f ∈ Fq[x] a polynomial of degree n ≥ 1, V(f) = #f(Fq) the number of different values f(α) of f, with α ∈ Fq, and p = q – V(f). It is shown that either ρ = 0 or 4n4 > q or 2pn > q. Hence, if q is “large” and f is not a permutation polynomial, then either n or ρ is “large”.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 43 , Issue 1 , February 1991 , pp. 141 - 146
Copyright
Copyright © Australian Mathematical Society 1991

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