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

Abstract

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 thatis 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.

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

“…where δ := m i=1 d i and D := m i=1 (d i − 1). Our approach relies on tools of algebraic geometry in the same vein as [CMPP14] and [MPP14]. In Section 2 we recall the basic notions and notations of algebraic geometry we use.…”

“…, Y s have full rank in A s . We remark that varieties defined by polynomials of this type arise in several combinatorial problems over finite fields (see, e.g., [CMP12], [CMPP14], [MPP14], [CMP15b] and [MPP15]). Finally, let A N := T d + a d−1 T d−1 + · · · + a 0 ∈ F q [T ] : G i (a d−1 , .…”

Polar varieties, Bertini's theorems and number of points of singular complete intersections over a finite field

“…Nevertheless, the exponentials in n in the second term of the right-hand side of (47) may hamper its application, even for low-dimensional varieties. In fact, in [10] and [28] we use (41) and (45) to estimate the average cardinality of the value set of polynomials with prescribed coefficients, with a significant gain over what is obtained applying (43) and (47).…”

Explicit estimates for polynomial systems defining irreducible smooth complete intersections

“…Let F q be the finite field with q elements, where q is a prime power. Multivariate polynomial systems over F q arise in connection with many fundamental problems in cryptography, coding theory, or combinatorics; see, e.g., Wolf & Preneel (2005), Ding et al (2006), Cafure et al (2012), Cesaratto et al (2014), . A random multivariate polynomial system over F q with more equations than variables is likely to be unsolvable over F q .…”