Generating series for irreducible polynomials over finite fields

We count the number of irreducible polynomials in several variables of a given degree over a finite field. The results are expressed in terms of a generating series, an exact formula and an asymptotic approximation. We also consider the case of the multi-degree and the case of indecomposable polynomials

Integral points on generic fibers

Let P(x,y) be a rational polynomial and k in Q be a generic value. If the curve (P(x,y)=k) is irreducible and admits an infinite number of points whose coordinates are integers then there exist algebraic automorphisms that send P(x,y) to the polynomial x or to x^2-dy^2. Moreover for such curves (and others) we give a sharp bound for the number of integral points (x,y) with x and y bounded.Comment: 12 page

Classification of polynomials from C^2 to C with one critical value

We give the classification, up to homeomorphisms, of reduced complex polynomials with 2 variables with one critical value.Comment: 17 pages, LaTeX2e, 2 figures, 2 tabulars. To appear in Mathematische Zeitscrift. One remark changed after theorem

Computation of Milnor numbers and critical values at infinity

We describe how to compute topological objects associated to a polynomial map of several complex variables with isolated singularities. These objects are: the affine critical values, the affine Milnor numbers for all irregular fibers, the critical values at infinity, and the Milnor numbers at infinity for all irregular fibers. Then for a family of polynomials we detect parameters where the topology of the polynomials can change. Implementation and examples are given with the computer algebra system Singular.Comment: 9 pages.To download the libraries for Singular see http://www-gat.univ-lille1.fr/~bodin

Non reality and non connectivity of complex polynomials

Using the same method we provide negative answers to the following questions: Is it possible to find real equations for complex polynomials in two variables up to topological equivalence (Lee Rudolph) ? Can two topologically equivalent polynomials be connected by a continuous family of topologically equivalent polynomials ?Comment: 6 pages, to appear in C. R. Acad. Sci. Pari