Pietro Corvaja

University of Udine, Udine, Friuli Venezia Giulia, Italy

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Publications (34)22.34 Total impact

  • Pietro Corvaja, Umberto Zannier
    Journal of differential geometry 03/2013; 93(3). · 1.18 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 S-integral relative to f^n(w) is finite and effectively computable. This may be thought of as a two-parameter 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 well-known theorem of Vojta.
    01/2012;
  • Pietro Corvaja, Umberto Zannier
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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 08/2011; 284(13):1652 - 1657. · 0.58 Impact Factor
  • Pietro Corvaja, Umberto Zannier
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    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 Brownawell-Masser 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 k-tuples”. 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). · 0.45 Impact Factor
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    Pietro Corvaja, Umberto Zannier
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    ABSTRACT: Minor technical changes. Section 4 improved. Comment: 27 pages, Plain TeX
    Asian Journal of Mathematics 01/2010; 14(2010). · 0.60 Impact Factor
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    Pietro Corvaja, Umberto Zannier
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    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 Zariski-dense. 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 01/2010; · 1.37 Impact Factor
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    Pietro Corvaja, Junjiro Noguchi
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    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 Corrales-Rodrigáñez 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 pulled-back divisor f * D of an ample divisor on an abelian variety A by an algebraically non-degenerate 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 semi-abelian varieties; namely, given two polarized semi-abelian varieties (A 1, D 1), (A 2, D 2) and algebraically non-degenerate 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; · 1.38 Impact Factor
  • Pietro Corvaja, Aaron Levin, Umberto Zannier
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    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). · 0.59 Impact Factor
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    Pietro Corvaja, Carlo Petronio, Umberto Zannier
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    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 Fermat-Euler Theorem on the representation of a prime as a sum of two squares.
    Elemente der Mathematik 11/2008;
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    ABSTRACT: We identify a new class of decidable hybrid automata: namely, parallel compositions of semi-algebraic o-minimal 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 semi-algebraic o-minimal 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 first-order logic over (﾿, 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 20-23, 2008. Proceedings; 01/2008
  • Pietro Corvaja, Umberto Zannier
    Rendiconti Lincei-matematica E Applicazioni - REND LINCEI-MAT APPL. 01/2008;
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    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 semi-algebraic o-minimal systems. The class of semi-algebraic o-minimal automata is not even closed under composition, i.e., the product of two automata of this class is not necessarily a semi-algebraic o-minimal automaton. However, we can prove our decidability result combining the decidability of both semi-algebraic 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 668-671;
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    Hybrid Systems: Computation and Control, 10th International Workshop, HSCC 2007, Pisa, Italy, April 3-5, 2007, Proceedings; 01/2007
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    Pietro Corvaja
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    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 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.
    11/2006;
  • Pietro Corvaja, U. Zannier
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    ABSTRACT: The present note is an addendum to our previous article "On a general Thue's equation." Corollary 2 in that paper provides a lower bound for the values of a polynomial with algebraic coefficients at integral points. Our purpose here is to prove a stronger version of it; also, we slightly simplify its proof, avoiding the use of Theorem 3 in the original paper.
    American Journal of Mathematics. 01/2006; 128(4):1057-1066.
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    Pietro Corvaja, Umberto Zannier
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    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; 2006:1-21. · 1.12 Impact Factor
  • Umberto Zannier, Pietro Corvaja
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    ABSTRACT: Markov pairs are pairs of coordinates from some solution (x,y,z) in positive integers of the equation x 2 +y 2 +z 2 =3xyz· The authors prove that the greatest prime factor of xy tends to infinity with max(x,y,z).
    Rendiconti del Seminario matematico della Università di Padova 01/2006; · 0.32 Impact Factor
  • Pietro Corvaja, Umberto Zannier
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    ABSTRACT: Minor technical changes. Section 4 improved.
    Journal of Algebraic Geometry. 12/2005; 17(2).
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    Pietro Corvaja, Umberto Zannier
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    ABSTRACT: In analogy with algebraic equations with S-units, we shall deal with S-unit points in an analytic hypersurface, or more generally with values of analytic functions at S-unit points.After proving a general theorem, we shall give diophantine applications to certain problems of integral points on subvarieties of . Also, we shall prove an analogue of a theorem of Masser, important in Mahler's method for transcendence.In the course of the proofs we shall also develop a theory for those algebraic subgroups of whose Zariski closure in An contains the origin. Among others, we shall prove a structure theorem for the family of such subgroups contained in a given analytic hypersurface, obtaining conclusions similar to the case of algebraic varieties.RésuméEn analogie avec les équations algébriques en S-unités, on considère ici des points S-unités sur une hypersurface analytique. On dérive d'un énoncé général des applications à des problèmes sur les points entiers des sous-variétés de . En plus, on déduit un analogue d'un lemme de zéros de Masser, qui intervient dans la méthode de Mahler en transcendance.Lors de la démonstration de ces résultats, on développe une théorie des sous-groupes algébriques de dont la clôture de Zariski dans An contient l'origine. En particulier, on démontre un théorème de structure pour la famille de tels sous-groupes contenus dans une hypersurface analytique, en obtenant une conclusion analogue à celle du cas des variétés algébriques.
    Annales Scientifiques de l’École Normale Supérieure. 01/2005;
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    Pietro Corvaja, Umberto Zannier
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    ABSTRACT: About fifty years ago Mahler proved that if $\alpha>1$ is rational but not an integer and if $0<l<1$ then the fractional part of $\alpha^n$ is $>l^n$ apart from a finite set of integers $n$ depending on $\alpha$ and $l$. Answering completely a question of Mahler we show that the same conclusion holds for all algebraic numbers which are not $d$-th roots of Pisot numbers. By related methods, we also answer a question by Mendes France, characterizing completely the quadratic irrationals $\alpha$ such that the continued fraction of $\alpha^n$ has period length tending to infinity.
    Acta Mathematica 03/2004; 193(2):175-191. · 2.71 Impact Factor