On the value set of small families of polynomials over a finite field, I

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:124

Article

Cesaratto, Eda^1,2  Matera, Guillermo^1,2  Perez, Mariana²  Privitelli, Melina^1,3 

[1] Univ Nacl Gen Sarmiento, Inst Desarrollo Humano, Buenos Aires, DF, Argentina

[2] Consejo Nacl Invest Cient & Tecn, Natl Council Sci & Technol, RA-1033 Buenos Aires, DF, Argentina

[3] Univ Nacl Gen Sarmiento, Inst Ciencias, Buenos Aires, DF, Argentina

关键词: Finite fields;    Average value set;    Symmetric polynomials;    Singular complete intersections;    Rational points;   

DOI  :  10.1016/j.jcta.2014.01.009

来源: Elsevier

【 摘 要 】

We obtain an estimate on the average cardinality of the value set of any family of monic polynomials of F-q[T] of degree d for which s consecutive coefficients a(d-1),...,a(d-s) are fixed. Our estimate holds without restrictions on the characteristic of F-q and asserts that nu(d, s, a) = mu dq + O(1), where nu(d, s, a) is such an average cardinality, mu(d) := Sigma(d)(r=1) (-1)(r-1)/r! and a := (a(d-1),..., a(d-s)). We provide an explicit upper bound for the constant underlying the O-notation in terms of d and s with good behavior. Our approach reduces the question to estimate the number of F-q-rational points with pairwisedistinct coordinates of a certain family of complete intersections defined over F-q. We show that the polynomials defining such complete intersections are invariant under the action of the symmetric group of permutations of the coordinates. This allows us to obtain critical information concerning the singular locus of the varieties under consideration, from which a suitable estimate on the number of F-q-rational points is established. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

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