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

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

Ivan\u27s Letter (Part 1)

The following cipher puzzle appeared in the May 1930 issue of The Enigma, the official publication of the National Puzzlers League. Erik Bodin offered a $10 to the first person to discover the secret message in Ivan\u27s letter, hinting only that the letter encoded the name of a point to be attacked, the date of the attack, and the troops involved . The cipher is unquestionably difficult; according to a brief not in the October 1930 Enigma, no one ever solved the puzzle. In the original article, the letter is presented in handwritten form; the slightly modified typewritten version given below preserves (and, in fact, makes somewhat easier to detect) the hidden message. The second half of the article, giving the solution to the cipher, will appear in the next issue of Word Ways

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