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

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

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