The number of reducible space curves over a finite field

Most hypersurfaces in projective space are irreducible, and rather precise estimates are known for the probability that a random hypersurface over a finite field is reducible. This paper considers the parametrization of space curves by the appropriate Chow variety, and provides bounds on the probability that a random curve over a finite field is reducible.

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

We obtain an estimate on the average cardinality of the value set of any family of monic polynomials of Fq[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 Fq and asserts that V(d,s,bfs{a})=mu_d.q+mathcal{O}(1), where V(d,s,bfs{a}) is such an average cardinality, mu_d:=sum_{r=1}^d{(-1)^{r-1}}/{r!} and bfs{a}:=(a_{d-1},.., d_{d-s}). We provide an explicit upper bound for the constant underlying the mathcal{O}--notation in terms of d and s with good behavior. Our approach reduces the question to estimate the number of Fq--rational points with pairwise--distinct coordinates of a certain family of complete intersections defined over Fq. 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 Fq--rational points is established.