Publications (43)33.39 Total impact
 Journal of differential geometry 03/2013; 93(3). · 1.09 Impact Factor
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ABSTRACT: The main result of this work is: “there are only finitely many odd perfect powers in ℕ having precisely four nonzero digits in their binary expansion”. The proof combines several tools: consequences of the Subspace Theorem (concerning integer values of analytic series), a general version of Roth’s theorem, Padé approxiamations in the 2adic case and Baker’s theory (lower bounds for linear forms in two logarithms in the archimedian and in the 2–adic cases).Annales Institut Fourier 01/2013; 2(2). DOI:10.5802/aif.2774 · 0.64 Impact Factor  01/2013; 59(3):225269. DOI:10.4171/LEM/5932

Article: Algebraic hyperbolicity of ramified covers of 𝔾 m 2 (and integral points on affine subsets of ℙ 2 )
Journal of differential geometry 01/2013; 93(3). · 1.09 Impact Factor  Journal of the European Mathematical Society 01/2013; 15(5):19271942. DOI:10.4171/JEMS/409 · 1.42 Impact Factor
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ABSTRACT: Let K be a number field, let f: P_1 > P_1 be a nonconstant rational map of degree greater than 1, let S be a finite set of places of K, and suppose that u, w in P_1(K) are not preperiodic under f. We prove that the set of (m,n) in N^2 such that f^m(u) is Sintegral relative to f^n(w) is finite and effectively computable. This may be thought of as a twoparameter analog of a result of Silverman on integral points in orbits of rational maps. This issue can be translated in terms of integral points on an open subset of P_1^2; then one can apply a modern version of the method of Runge, after increasing the number of components at infinity by iterating the rational map. Alternatively, an ineffective result comes from a wellknown theorem of Vojta.Journal für die reine und angewandte Mathematik (Crelles Journal) 01/2012; DOI:10.1515/crelle20130060 · 1.30 Impact Factor 
Article: On the rank of certain matrices
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ABSTRACT: This note is an appendix to the paper “Osculating spaces and diophantine equations (with an Appendix by Pietro Corvaja and Umberto Zannier)” by M. Bolognesi and G. Pirola.Mathematische Nachrichten 09/2011; 284(13):1652  1657. DOI:10.1002/mana.200810300 · 0.66 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: In the paper under review, the authors give a lower bound for the number of distinct zeros of the sum 1+u+v, where u and v are rational functions which is sharp when u and v have few distinct zeros and poles compared to their degrees. Their main result sharpens the “abcd” theorem of BrownawellMasser and Voloch in some cases which are sufficient to obtain new finiteness results for Diophantine equations with polynomials. The main tool is a prior result of the authors from [“Some cases on Vojta’s conjecture on integral points over finite fields”, J. Algebr. Geom. 17, No. 2, 295–333 (2008); addendum Asian J. Math. 14, No. 4, 581–584 (2010; Zbl 1221.11146)]. As applications, they obtain that the Fermat surface x a +y b +z c =1 contains only finitely many rational or elliptic curves when a≥10 4 and c≥2. They also obtain an interesting application to the so–called “Diophantine ktuples”. Namely, if a,b,c are three distinct nonzero complex polynomials not all constant such that 1+ab=x p ,1+ac=y q and 1+bc=z r with complex polynomials x,y,z and integers p,q,r≥864, then after suitably permuting a,b,c, we have c 2 +1=0 and a+b=2c.Bulletin de la Société mathématique de France 01/2011; 139(4). · 0.52 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We prove some new degeneracy results for integral points and entire curves on surfaces; in particular, we provide the first examples, to our knowledge, of a simply connected smooth variety whose sets of integral points are never Zariskidense. 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.Advances in Mathematics 10/2010; 225(2225):10951118. DOI:10.1016/j.aim.2010.03.017 · 1.35 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: Minor technical changes. Section 4 improved. Comment: 27 pages, Plain TeXAsian Journal of Mathematics 01/2010; 14(2010). DOI:10.4310/AJM.2010.v14.n4.a4 · 0.42 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: In 1988 Erdös asked if the prime divisors of x n − 1 for all n = 1, 2, … determine the given integer x; the problem was affirmatively answered by CorralesRodrigáñez and Schoof (J Number Theory 64:276–290, 1997) [but a solution could also be deduced from an earlier result of Schinzel (Bull Acad Polon Sci 8:307–309, 2007)] together with its elliptic version. Analogously, Yamanoi (Forum Math 16:749–788, 2004) proved that the support of the pulledback divisor f * D of an ample divisor on an abelian variety A by an algebraically nondegenerate entire holomorphic curve f : C → A essentially determines the pair (A, D). By making use of the main theorem of Noguchi (Forum Math 20:469–503, 2008) we here deal with this problem for semiabelian varieties; namely, given two polarized semiabelian varieties (A 1, D 1), (A 2, D 2) and algebraically nondegenerate entire holomorphic curves f i : C → A i , i=1, 2, we classify the cases when the inclusion Suppf1*D1 Ì Suppf2* D2{{\rm{Supp}}\, f_1^*D_1\subset {\rm{Supp}}\, f_2^* D_2} holds. We shall remark in §5 that these methods yield an affirmative answer to a question of Lang formulated in 1966. Our answer is more general and more geometric than the original question. Finally, we interpret the main result of Corvaja and Zannier (Invent Math 149:431–451, 2002) to provide an arithmetic counterpart in the toric case.Mathematische Annalen 07/2009; DOI:10.1007/s002080110692x · 1.20 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We prove sufficient conditions for the degeneracy of integral points on certain threefolds and other varieties of higher dimension. In particular, under a normal crossings assumption, we prove the degeneracy of integral points on an affine threefold with seven ample divisors at infinity. Analogous results are given for holomorphic curves. As in our previous works [2], [5], the main tool involved is Schmidt's Subspace Theorem, but here we introduce a technical novelty which leads to stronger results in dimension three or higher.Tohoku Mathematical Journal 01/2009; 61(2009). DOI:10.2748/tmj/1264084501 · 0.60 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We consider the question of existence of ramified covers over P_1 matching certain prescribed ramification conditions. This problem has already been faced in a number of papers, but we discuss alternative approaches for an existence proof, involving elliptic curves and universal ramified covers with signature. We also relate the geometric problem with finite permutation groups and with the FermatEuler Theorem on the representation of a prime as a sum of two squares.Elemente der Mathematik 11/2008; DOI:10.4171/EM/207  [Show abstract] [Hide abstract]
ABSTRACT: The authors give an overview (Theorems A–L) of their recent results on questions about linear recurrences and Sintegral points on affine varieties, for which they used the Subspace Theorem. To illustrate the central idea of their proofs they show the following claim: if for integers 1<b<a one has a n 1 b n 1∈ℤ for infinitely many n∈ℕ then a is a power of b. As an illustrative example for their geometric results the authors sketch the proof of the following claim: given an affine part of a nonsingular projective curve over a number field such that the divisor at infinity has at least 3 different points, then this affine part contains only finitely many Sintegral points.  [Show abstract] [Hide abstract]
ABSTRACT: Let C be an absolutely irreducible curve in 𝔾 m n of genus g and degree d, being defined over some number field k. Let r be the minimal dimension of a subgroup H of 𝔾 m n such that C is contained in a translate of H by a torsion point, and suppose that r≥2. The authors prove that the order m of any torsion point of C satisfies the inequality φ(m) 3 m 2/r ≤108(r!) 2/r [k:ℚ] 3 d 2 (g1+rd), which yields an upper bound for m since φ(m)≫m/loglogm. For n=r=2 this specializes to m(loglogm) 3/2 ≪[k:ℚ] 3/2 dd+g· The novelty of this result is that this upper bound contains the genus of the curve and thus improves (for small genus) the known upper bound of order of magnitude d 2 .Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica E Applicazioni 01/2008; DOI:10.4171/RLM/509 · 0.68 Impact Factor 
Conference Paper: Decidable Compositions of OMinimal Automata.
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ABSTRACT: We identify a new class of decidable hybrid automata: namely, parallel compositions of semialgebraic ominimal automata. The class we consider is fundamental to hierarchical modeling in many exemplar systems, both natural and engineered. Unfortunately, parallel composition, which is an atomic operator in such constructions, does not preserve the decidability of reachability. Luckily, this paper is able to show that when one focuses on the composition of semialgebraic ominimal automata, it is possible to translate the decidability problem into a satisfiability problem over formulæinvolving both real and integer variables. While in the general case such formulæ would be undecidable, the particular format of the formulæ obtained in our translation allows combining decidability results stemming from both algebraic number theory and firstorder logic over (ï¾¿, 0, 1, + , *, < ) to yield a novel decidability algorithm. From a more general perspective, this paper exposes many new open questions about decidable combinations of real/integer logics.Automated Technology for Verification and Analysis, 6th International Symposium, ATVA 2008, Seoul, Korea, October 2023, 2008. Proceedings; 01/2008  [Show abstract] [Hide abstract]
ABSTRACT: This paper addresses questions regarding the decidability of hybrid automata that may be constructed hierarchically and in a modular way, as is the case in many exemplar systems, be it natural or engineered. Since an important step in such constructions is a product operation, which constructs a new product hybrid automaton by combining two simpler component hybrid automata, an essential property that would be desired is that the reachability property of the product hybrid automaton be decidable, provided that the component hybrid automata belong to a suitably restricted family of automata. Somewhat surprisingly, the product operation does not assure a closure of decidability for the reachability problem. Nonetheless, this paper establishes the decidability of the reachability condition over automata which are obtained by composing two semialgebraic ominimal systems. The class of semialgebraic ominimal automata is not even closed under composition, i.e., the product of two automata of this class is not necessarily a semialgebraic ominimal automaton. However, we can prove our decidability result combining the decidability of both semialgebraic formulæ over the reals and linear Diophantine equations. All the proofs of the results presented in this paper can be found in [1].05/2007: pages 668671; 
Conference Paper: Composing Semialgebraic OMinimal Automata.
Hybrid Systems: Computation and Control, 10th International Workshop, HSCC 2007, Pisa, Italy, April 35, 2007, Proceedings; 01/2007  [Show abstract] [Hide abstract]
ABSTRACT: 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 Zariskidense 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 byproduct of the proof, we obtain a version of the classical Hilbert Irreducibility Theorem valid for linear algebraic groups.  [Show abstract] [Hide abstract]
ABSTRACT: We study integral points on affine surfaces by means of a new method, relying on the Subspace Theorem. Under suitable assumptions on the divisor at infinity, we prove that the integral points are contained in a curve. As a corollary, we prove that there are only finitely many quadratic integral points on an affine curve with five points at infinity.International Mathematics Research Notices 01/2006; 2006:121. DOI:10.1155/IMRN/2006/98623 · 1.07 Impact Factor
Publication Stats
255  Citations  
33.39  Total Impact Points  
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Institutions

1997–2011

University of Udine
 Department of Mathematical and Computer Science
Udine, Friuli Venezia Giulia, Italy


2005

Scuola Normale Superiore di Pisa
Pisa, Tuscany, Italy
