A Note on Value Sets of Polynomials over Finite Fields

2017

A classical result on value sets of non-permutation polynomials over finite fields is due to Wan (1993). Denoting the cardinality of the value set of f 2 Fq[x] by jVf j, Wan's result gives the upper bound JVx, where d is the degree of f. A proof of this bound due to Turnwald, which was obtained by the use of symmetric polynomials is given in Chapter 2. A generalization of this result was obtained by Aitken that we also describe here. The work of Aitken focuses on value sets of pairs of polynomials in Fq[x], in particular, he studies the size of the intersection of their value sets. We present pairs of particular polynomials whose value sets do not only have the same size but are actually identical. Clearly, a permutation polynomial f of Fq[x] satisfies jVf j = q. In Chapter 3, we discuss permutation behaviour of pairs of polynomials in Fq[x].

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 V(d, s, a) = µ d q + O(1), where V(d, s, a) is such an average cardinality, µ d := d r=1 (−1) r−1 /r! and a := (a d−1 ,. .. , d 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 pairwise-distinct 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.

Proceedings of the American Mathematical Society, 1993

Let ¥q be the finite field of q elements, and let Vf be the number of values taken by a polynomial f{x) over ¥q. We establish a lower bound and an upper bound of Vf in terms of certain invariants of f(x). These bounds improve and generalize some of the previously known bounds of Vf. In particular, the classical Hermite-Dickson criterion is improved. Our bounds also give a new proof of a recent theorem of Evans, Greene, and Niederreiter. Finally, we give some examples which show that our bounds are sharp.

Journal of Combinatorial Theory, Series A, 2014

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 V(d, s, a) = µ d q + O(1), where V(d, s, a) is such an average cardinality, µ d := d r=1 (−1) r−1 /r! and a := (a d−1 ,. .. , d 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 pairwise-distinct 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.

Acta Arithmetica, 2014

arXiv (Cornell University), 2015

We estimate the average cardinality V(A) of the value set of a general family A of monic univariate polynomials of degree d with coefficients in the finite field Fq. We establish conditions on the family A under which V(A) = µ d q + O(q 1/2), where µ d := d r=1 (−1) r−1 /r!. The result holds without any restriction on the characteristic of Fq and provides an explicit expression for the constant underlying the O-notation in terms of d. We reduce the question to estimating the number of Fq-rational points with pairwise-distinct coordinates of a certain family of complete intersections defined over Fq. For this purpose, we obtain an upper bound on the dimension of the singular locus of the complete intersections under consideration, which allows us to estimate the corresponding number of Fq-rational points.

Finite Fields and Their Applications, 1996

Finite Fields and Their Applications, 2011

In this paper we study the relation between coefficients of a polynomial over finite field F q and the moved elements by the mapping that induces the polynomial. The relation is established by a special system of linear equations. Using this relation we give the lower bound on the number of nonzero coefficients of polynomial that depends on the number m of moved elements. Moreover we show that there exist permutation polynomials of special form that achieve this bound when m | q − 1. In the other direction, we show that if the number of moved elements is small then there is an recurrence relation among these coefficients. Using these recurrence relations, we improve the lower bound of nonzero coefficients when m q−1 and m ≤ q−1 2. As a byproduct, we show that the moved elements must satisfy certain polynomial equations if the mapping induces a polynomial such that there are only two nonzero coefficients out of 2m consecutive coefficients. Finally we provide an algorithm to compute the coefficients of the polynomial induced by a given mapping with O(q 3/2) operations.

Contemporary Developments in Finite Fields and Applications, 2016

For any given polynomial f over the finite field Fq with degree at most q − 1, we associate it with a q × q matrix A(f) = (a ik) consisting of coefficients of its powers (f (x)) k = q−1 i=0 a ik x i modulo x q − x for k = 0, 1,. .. , q − 1. This matrix has some interesting properties such as A(g • f) = A(f)A(g) where (g • f)(x) = g(f (x)) is the composition of the polynomial g with the polynomial f. In particular, A(f (k)) = (A(f)) k for any k-th composition f (k) of f modulo x q − x with k ≥ 0. As a consequence, we prove that the rank of A(f) gives the cardinality of the value set of f. Moreover, if f is a permutation polynomial then the matrix associated with its inverse A(f (−1)) = A(f) −1 = P A(f)P where P is an antidiagonal permutation matrix. As an application, we study the period of a nonlinear congruential pseduorandom sequenceā = {a 0 , a 1 , a 2 , ...} generated by an = f (n) (a 0) with initial value a 0 , in terms of the order of the associated matrix. Finally we show that A(f) is diagonalizable in some extension field of Fq when f is a permutation polynomial over Fq. v f = (a 0 , a 1 ,. .. , a q−1) T .

Finite Fields and Their Applications, 2022

In this paper we introduce the additive analogue of the index of a polynomial over finite fields. We study several problems in the theory of polynomials over finite fields in terms of their additive indices, such as value set sizes, bounds on multiplicative character sums, and characterizations of permutation polynomials.