Composite lacunary polynomials and the proof of a conjecture of Schinzel

Let $g(x)$ be a fixed non-constant complex polynomial. It was conjectured by Schinzel that if $g(h(x))$ has boundedly many terms, then h(x)\in \C[x] must also have boundedly many terms. Solving an older conjecture raised by R\'enyi and by Erd\"os, Schinzel had proved this in the special cases $g(x)=x^d$; however that method does not extend to the general case. Here we prove the full Schinzel's conjecture (actually in sharper form) by a completely different method. Simultaneously we establish an "algorithmic" parametric description of the general decomposition $f(x)=g(h(x))$, where $f$ is a polynomial with a given number of terms and $g,h$ are arbitrary polynomials. As a corollary, this implies for instance that a polynomial with $l$ terms and given coefficients is non-trivially decomposable if and only if the degree-vector lies in the union of certain finitely many subgroups of $\Z^l$.Comment: 9 page

Hilbert Irreducibility above algberaic groups

The paper offers versions of Hilbert's Irreducibility Theorem for the lifting of points in a cyclic subgroup of an algebraic group to a ramified cover. A version of Bertini Theorem in this context is also obtained.Comment: 22 page

On the Hilbert Property and the Fundamental Group of Algebraic Varieties

We review, under a perspective which appears different from previous ones, the so-called Hilbert Property (HP) for an algebraic variety (over a number field); this is linked to Hilbert's Irreducibility Theorem and has important implications, for instance towards the Inverse Galois Problem. We shall observe that the HP is in a sense `opposite' to the Chevalley-Weil Theorem, which concerns unramified covers; this link shall immediately entail the result that the HP can possibly hold only for simply connected varieties (in the appropriate sense). In turn, this leads to new counterexamples to the HP, involving Enriques surfaces. We also prove the HP for a K3 surface related to the above Enriques surface, providing what appears to be the first example of a non-rational variety for which the HP can be proved. We also formulate some general conjectures relating the HP with the topology of algebraic varieties.Comment: 24 page

Rational points in periodic analytic sets and the Manin-Mumford conjecture

We present a new proof of the Manin-Mumford conjecture about torsion points on algebraic subvarieties of abelian varieties. Our principle, which admits other applications, is to view torsion points as rational points on a complex torus and then compare (i) upper bounds for the number of rational points on a transcendental analytic variety (Bombieri-Pila-Wilkie) and (ii) lower bounds for the degree of a torsion point (Masser), after taking conjugates. In order to be able to deal with (i), we discuss (Thm. 2.1) the semi-algebraic curves contained in an analytic variety supposed invariant for translations by a full lattice, which is a topic with some independent motivation.Comment: 12 page

A lower bound for the height of a rational function at SS-unit points

Let $\Gamma$ be a finitely generated subgroup of the multiplicative group \G_m^2(\bar{Q}). Let p(X,Y),q(X,Y)\in\bat{Q} be two coprime polynomials not both vanishing at $(0,0)$; let $\epsilon>0$. We prove that, for all $(u,v)\in\Gamma$ outside a proper Zariski closed subset of $G_m^2$, the height of $p(u,v)/q(u,v)$ verifies $h(p(u,v)/q(u,v))>h(1:p(u,v):q(u,v))-\epsilon \max(h(uu),h(v))$. As a consequence, we deduce upper bounds for (a generalized notion of) the g.c.d. of $u-1,v-1$ for $u,v$ running over $\Gamma$.Comment: Plain TeX 18 pages. Version 2; minor changes. To appear on Monatshefte fuer Mathemati

Integral points, divisibility between values of polynomials and entire curves on surfaces

We prove some new degeneracy results for integral points and entire curves on surfaces; in particular, we provide the first example, to our knowledge, of a simply connected smooth variety whose sets of integral points are never Zariski-dense (and no entire curve has Zariski-dense image). Some of our results are connected with divisibility problems, i.e. the problem of describing the integral points in the plane where the values of some given polynomials in two variables divide the values of other given polynomials.Comment: minor changes, two references adde