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

Abstract

Abstract. We obtain an estimate on the average cardinality of the value set of any family of monic polynomials ofis the average second moment on any family of monic polynomials of F q [T ] of degree d with s consecutive coefficients fixed as above. Finally, we show that, where V 2 (d, 0) denotes the average second moment of all monic polynomials in F q [T ] of degree d with f (0) = 0. All our estimates hold for fields of characteristic p > 2 and provide explicit upper bounds for the constants underlying the O-notation in terms of d and s with "good" behavior. Our approach reduces the questions to estimate the number of F q -rational points with pairwise-distinct coordinates of a certain family of complete intersections defined over F q . A critical point for our results is an analysis of the singular locus of the varieties under consideration, which allows to obtain rather precise estimates on the corresponding number of F q -rational points.

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).…”

The distribution of factorization patterns on linear families of polynomials over a finite field

“…To this end, observe that a polynomial f ∈ A is not square-free if and only if its discriminant is equal to zero. In [MPP13] we study the so-called discriminant locus of A, namely the set A nsq formed by the elements of A whose discriminant is equal to zero (see also [FS84] for further results on discriminant loci). According to [MPP13,Theorem A.3], the discriminant locus A nsq is the set of F q -rational points of a hypersurface of degree n(n−1) of a suitable (n − m)-dimensional affine space.…”

Explicit estimates for the number of rational points of singular complete intersections over a finite field

“…Let A nsq be discriminant locus of A, i.e., the set of elements of A whose discriminant is equal to zero. In [FS84] and [MPP14] discriminant loci are studied. In particular, from [FS84] one easily deduces that the discriminant locus A nsq is the set of F q -rational points of a hypersurface of degree at most n(n − 1) of a suitable (n − m)-dimensional affine space.…”