Genus one factors of curves defined by separated variable polynomials

We give some sufficient conditions on complex polynomials P and Q to assure that the algebraic plane curve P(x)-Q(y)=0 has no irreducible component of genus 0 or 1. Moreover, if deg (P)=deg (Q) and if both P, Q satisfy Hypothesis I introduced by H. Fujimoto, our sufficient conditions are necessary

Algebraic Degeneracy of Non-Archimedean Analytic Maps

We prove non-Archimedean analogs of results of Noguchi and Winkelmann showing algebraic degeneracy of rigid analytic maps to projective varieties omitting an effective divisor with sufficiently many irreducible components relative to the rank of the group they generate in the Neron-Severi group of the variety.Comment: 10 page

A formal proof of the Kepler conjecture

This article describes a formal proof of the Kepler conjecture on dense sphere packings in a combination of the HOL Light and Isabelle proof assistants. This paper constitutes the official published account of the now completed Flyspeck project

New applications of the p-adic Nevanlinna Theory

International audienceLet IK be an algebraically closed field of characteristic 0 complete for an ultrametric absolute value. Following results obtained in complex analysis , here we examine problems of uniqueness for meromorphic functions having finitely many poles, sharing points or a pair of sets (C.M. or I.M.) defined either in the whole field IK or in an open disk, or in the complement of an open disk. Following previous works in l C, we consider functions f n (x)f m (ax + b), g n (x)g m (ax + b) with |a| = 1 and n = m, sharing a rational function and we show that f g is a n + m-th root of 1 whenever n + m ≥ 5. Next, given a small function w, if n, m ∈ IN are such that |n−m|∞ ≥ 5, then f n (x)f m (ax+b)−w has infinitely many zeros. Finally, we examine branched values for meromorphic functions f n (x)f m (ax + b)