Hostname: page-component-5db58dd55d-8mwbx Total loading time: 0 Render date: 2026-06-07T15:06:47.498Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

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”.

Information

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

References

[1]Aho, A.V., Hopcroft, J.E. and Ullman, J.D., The design and analysis of computer algorithms (Addison-Wesley, Reading, MA, 1974).Google Scholar
[2]Bombieri, E. and Davenport, H., ‘On two problems of Mordell’, Amer. J. Math. 88 (1966), 61–70.CrossRefGoogle Scholar
[3]Cohen, S.D., ‘The distribution of polynomials over finite fields’, Acta Arith. 17 (1970), 255–271.CrossRefGoogle Scholar
[4]Davenport, H. and Lewis, D.J., ‘Notes on congruences (I)’, Quart. J. Math. Oxford 14 (1963), 51–60.CrossRefGoogle Scholar
[5]von zur Gathen, J., ‘Tests for permutation polynomials’, SIAM J. Comput. (to appear).Google Scholar
[6]von zur Gathen, J. and Kaltofen, E., ‘Factorization of multivariate polynomials over finite fields’, Math. Comp. 45 (1985), 251–261.CrossRefGoogle Scholar
[7]Gwehenberger, G., Über die Darstellung von Permutationen durch Polynome und rationale Funktionen, PhD thesis (TH Wien, 1970).Google Scholar
[8]Hayes, D.R., ‘A geometric approach to permutation polynomials over a finite field’, Duke Math. J. 34 (1967), 293–305.CrossRefGoogle Scholar
[9]Kaltofen, E., ‘Fast parallel absolute irreducibility testing’, J Symbolic Comput. 1 (1985), 57–67.CrossRefGoogle Scholar
[10]Lidl, R. and Mullen, G.L., ‘When does a polynomial over a finite field permute the elements of the field’, Amer. Math. Monthly 95 (1988), 243–246.CrossRefGoogle Scholar
[11]Lidl, R. and Niederreiter, H., Finite fields: Encyclopedia of Mathematics and its Applications 20 (Addison-Wesley, Reading MA, 1983).Google Scholar
[12]MacCluer, C.R., ‘On a conjecture of Davenport and Lewis concerning exceptional polynomials’, Acta Arith. 12 (1967), 289–299.CrossRefGoogle Scholar
[13]Tietäväinen, A., ‘On non-residues of a polynomial’, Ann. Univ. Turku Ser. A 94 (1966).Google Scholar
[14]Wan, D., ‘On a conjecture of Carlitz’, J. Austral. Math. Soc. (Series A) 43 (1987), 375–384.Google Scholar
[15]Williams, K.S., ‘On extremal polynomials’, Canad. Math. Bull. 10 (1967), 585–594.CrossRefGoogle Scholar
[16]Williams, K.S., ‘On exceptional polynomials’, Canad. Math. Bull. 11 (1968), 279–282.CrossRefGoogle Scholar