Rational fixed points for linear group actions

Let $k$ be a finitely generated field, let $X$ be an algebraic variety and $G$ a linear algebraic group, both defined over $k$. Suppose $G$ acts on $X$ and every element of a Zariski-dense semigroup $\Gamma \subset G(k)$ has a rational fixed point in $X(k)$. We then deduce, under some mild technical assumptions, the existence of a rational map $G\to X$, defined over $k$, sending each element $g\in G$ to a fixed point for $g$. The proof makes use of a recent result of Ferretti and Zannier on diophantine equations involving linear recurrences. As a by-product of the proof, we obtain a version of the classical Hilbert Irreducibility Theorem valid for linear algebraic groups.Comment: 35 pages, Plain Tex. A gap in the previous proof of Theorem 1.2 overcome, plus minor changes. Thanks to J. Bernik and the refere

Phase-noise limitations in continuous-variable quantum key distribution with homodyne detection

In continuous-variables quantum key distribution with coherent states, the advantage of performing the detection by using standard telecoms components is counterbalanced by the lack of a stable phase reference in homodyne detection due to the complexity of optical phase-locking circuits and to the unavoidable phase noise of lasers, which introduces a degradation on the achievable secure key rate. Pilot-assisted phase-noise estimation and postdetection compensation techniques are used to implement a protocol with coherent states where a local laser is employed and it is not locked to the received signal, but a postdetection phase correction is applied. Here the reduction of the secure key rate determined by the laser phase noise, for both individual and collective attacks, is analytically evaluated and a scheme of pilot-assisted phase estimation proposed, outlining the tradeoff in the system design between phase noise and spectral efficiency. The optimal modulation variance as a function of the phase-noise amount is derived

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

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