Value sets of polynomials over finite fields

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.

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.

Journal of Number Theory, 2013

We give an explicit characterization of all minimal value set polynomials in F q [x] whose set of values is a subfield F q ′ of F q . We show that the set of such polynomials, together with the constants of F q ′ , is an F q ′ -vector space of dimension 2 [Fq:F q ′ ] . Our approach not only provides the exact number of such polynomials, but also yields a construction of new examples of minimal value set polynomials for some other fixed value sets. In the latter case, we also derive a non-trivial lower bound for the number of such polynomials.

Canadian Journal of Mathematics, 1969

If A is a set with only a finite number of elements, we write |A| for the number of elements in A. Let p be a large prime and let m be a positive integer fixed independently of p. We write [pm ] for the finite field with pm elements and [pm ]′ for [pm ] – {0}. We consider in this paper only subsets H of [pm ] for which |H| = h satisfies 1.1 If f(x) ∈ [pm, x] we let N(f; H) denote the number of distinct values of y in H for which at least one of the roots of f(x) = y is in [pm ]. We write d(d ≥ 1) for the degree of f and suppose throughout that d is fixed and that p ≧ p 0(d), for some prime p 0, depending only on d, which is greater than d.

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.

Finite Fields and Their Applications, 2012

We show that, for any integer ℓ with q − √ p − 1 ≤ ℓ < q − 3 where q = p n and p > 9, there exists a multiset M satisfying that 0 ∈ M has the highest multiplicity ℓ and b∈M b = 0 such that every polynomial over finite fields Fq with the prescribed range M has degree greater than ℓ. This implies that Conjecture 5.1. in [1] is false over finite field Fq for p > 9 and k := q − ℓ − 1 ≥ 3.

2017

In this paper, assuming a conjecture of Vojta&#39;s on bounded degree algebraic numbers and a quantitative version of Northcott&#39;s theorem over number field $k$, we show the existence of explicit lower and upper bounds for the number of polynomials $f\in k[x]$ of degree $r$ whose irreducible factors have multiplicity strictly less than $s$ and moreover $f(b_1),\cdots, f(b_M)$ are $s$-powerful values for a certain integer $M$, where $b_i$&#39;s belong to a sequence of the pairwise distinct element of $k$ that satisfy certain conditions. Our results improve the recent work of H. Pasten on the subject.

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.

Linear Algebra and its Applications, 2020

Linearized polynomials appear in many different contexts, such as rank metric codes, cryptography and linear sets, and the main issue regards the characterization of the number of roots from their coefficients. Results of this type have been already proved in [7, 10, 24]. In this paper we provide bounds and characterizations on the number of roots of linearized polynomials of this form ax + b 0 x q s + b 1 x q s+n + b 2 x q s+2n +. .. + b t−1 x q s+n(t−1) ∈ F q nt [x], with gcd(s, n) = 1. Also, we characterize the number of roots of such polynomials directly from their coefficients, dealing with matrices which are much smaller than the relative Dickson matrices and the companion matrices used in the previous papers. Furthermore, we develop a method to find explicitly the roots of a such polynomial by finding the roots of a q n-polynomial. Finally, as an applications of the above results, we present a family of linear sets of the projective line whose points have a small spectrum of possible weights, containing most of the known families of scattered linear sets. In particular, we carefully study the linear sets in PG(1, q 6) presented in [9].

Acta Arithmetica, 2015

For an odd prime p and an integer w ≥ 1, polynomial quotients qp,w(u) are defined by

Linear Algebra and its Applications, 1990

We investigate the number of distinct values taken by a polynomial of a fixed degree over a finite field EC,. Exact formulae are obtained for the number of polynomials of degree > 9 taking on a specified number of distinct values and for the corresponding variance about the mean. This extends previous results of Cohen and Uchiyama, who derived formulae for the average number of distinct values of such polynomials. 1.

Journal of Pure and Applied Algebra, 2015

We determine upper bounds on the number of rational points of an affine or projective algebraic set defined over an extension of a finite field by a system of polynomial equations, including the case where the algebraic set is not defined over the finite field by itself. A special attention is given to irreducible but not absolutely irreducible algebraic sets, which satisfy better bounds. We study the case of complete intersections, for which we give a decomposition, coarser than the decomposition in irreducible components, but more directly related to the polynomials defining the algebraic set. We describe families of algebraic sets having the maximum number of rational points in the affine case, and a large number of points in the projective case. Nous déterminons des majorations du nombre de points d'un ensemble algébrique affine ou projectif, défini sur une extension d'un corps fini par un système d'équations polynomiales, y compris dans le cas où l'ensemble algébrique n'est pas défini sur le corps fini lui-même. Une attention particulière est portée aux ensemble algébriques irréductibles mais non absolument irréductibles, pour lesquels nous obtenons de meilleures bornes. Nous étudions le cas des intersections complètes, pour lesquelles nous construisons une décomposition moins fine que la décomposition en composantes irréductibles, mais plus directement liée aux polynômes qui définissent l'ensemble algébrique. Enfin, nous construisons des familles d'ensembles algébriques atteignant le nombre maximum de points rationnels dans le cas affine, et comportant de nombreux points dans le cas projectifs.

Moscow Mathematical Journal

We give a complete conjectural formula for the number er(d, m) of maximum possible Fq-rational points on a projective algebraic variety defined by r linearly independent homogeneous polynomial equations of degree d in m + 1 variables with coefficients in the finite field Fq with q elements, when d < q. It is shown that this formula holds in the affirmative for several values of r. In the general case, we give explicit lower and upper bounds for er(d, m) and show that they are sometimes attained. Our approach uses a relatively recent result, called the projective footprint bound, together with results from extremal combinatorics such as the Clements-Lindström Theorem and its variants. Applications to the problem of determining the generalized Hamming weights of projective Reed-Muller codes are also included.

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 .

The Ramanujan Journal, 2018

Let D be a finite subset of a commutative ring R with identity. Let f (x) ∈ R[x] be a polynomial of degree d. For a nonnegative integer k, we study the number In this paper, we establish several bounds for the difference between N f (D, k, b) and the expected main term 1 |R| |D| k , depending on the nature of the finite ring R and f . For R = Zn, let p = p(n) be the smallest prime divisor of n, answering an open question raised by Stanley [29] in a general setting, where . Furthermore, if n is a prime power, then δ(n) = 1/p and one can take C d = 4.41. Similar and stronger bounds are given for two more cases. The first one is when R = Fq, a q-element finite field of characteristic p and f (x) is genetic. The second one is essentially the well-known subset sum problem over an arbitrary finite abelian group. These bounds extend several previous results.