Yuri Bilu

Polynomials with roots modulo every integer

Given a polynomial with integer coefficients, we calculate the density of the set of primes modulo which the polynomial has a root. We also give a simple criterion to decide whether or not the polynomial has a root modulo every non-zero integer.

EFFECTIVE ANALYSIS OF INTEGRAL POINTS ON ALGEBRAIC CURVES

AbstractLetK be an algebraic number field,S⊇S\t8 a finite set of valuations andC a non-singular algebraic curve overK. Letx∈K(C) be non-constant. A pointP∈C(K) isS-integral if it is not a pole ofx and |x(P)|v>1 impliesv∈S. It is proved that allS-integral points can be effectively determined if the pair (C, x) satisfies certain conditions. In particular, this is the case if(i)x:C→P1 is a Galois covering andg(C)≥1;(ii)the integral closure of

Quantitative Siegel's theorem for Galois coverings

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Octahedrons with Equally Many Lattice Points

[x] in

Divisibility of Class Numbers: Enumerative Approach

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CHARACTERIZING ALGEBRAIC CURVES WITH INFINITELY MANY INTEGRAL POINTS

(C) has at least two units multiplicatively independent mod