
Journal of differential geometry 03/2013; 93(3). · 1.42 Impact Factor

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ABSTRACT: Let X be a smooth affine surface, X â G2m be a finite morphism. We study the affine curves on X, with bounded genus and number of points at infinity, obtaining bounds for their degree in terms of Euler characteristic. A typical example where these bounds hold is represented by the complement of a threecomponent curve in the projective plane, of total degree at least 4. The corresponding results may be interpreted as bounding the height of integral points on X over a function field. In the language of Diophantine Equations, our results may be rephrased in terms of bounding the height of the solutions of f(u, v, y) = 0, with u, v, y over a function field, u, v Sunits. It turns out that all of this contain some cases of a strong version of a conjecture of Vojta over function fields in the split case. Moreover, our method would apply also to the nonsplit case. We remark that special cases of our results in the holomorphic context were studied by M. Green already in the seventies, and recently in greater generality by Noguchi, Winkelmann, and Yamanoi [12]; however, the algebraic context was left open and seems not to fall in the existing techniques. Journal of differential geometry 03/2013; 93(3). · 1.42 Impact Factor

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ABSTRACT: We prove that there are only finitely many odd perfect powers in N having precisely four nonzero digits in their binary expansion. The proofs in fact lead to more general results, but we have preferred to limit ourselves to the present statement for the sake of simplicity and clarity of illustration of the methods. These methods combine various ingredients: results (derived from the Subspace Theorem) on integer values of analytic series at Sunit points (in a suitable adic convergence), Roth's general theorem, 2adic PadĂ© approximations (by integers) to numbers in varying number fields and lower bounds for linear forms in two logarithms (both in the usual and in the 2adic context). Annales Institut Fourier 01/2013; 2(2). DOI:10.5802/aif.2774 · 0.67 Impact Factor

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ABSTRACT: In our previous work [4] we proved a bound for gcd(u  1, v  1), for Sunits u, v of a function field in characteristic zero. This generalized an analogous bound holding over number fields, proved in [3]. As pointed out by Silverman [15], the exact analogue does not work for function fields in positive characteristic. In the present work, we investigate possible extensions in that direction; it turns out that under suitable assumptions some of the results still hold. For instance we prove Theorems 2 and 3 below, from which we deduce in particular a new proof of Weil's bound for the number of rational points on a curve over finite fields (see Â§4). When the genus of the curve is large compared to the characteristic, we can even go beyond it. What seems a new feature is the analogy with the characteristic zero case, which admitted applications to apparently distant problems. Journal of the European Mathematical Society 01/2013; 15(5):19271942. DOI:10.4171/JEMS/409 · 1.70 Impact Factor

01/2013; 59(3):225269. DOI:10.4171/LEM/5932

Source Available from: Thomas J Tucker
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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 fĂŒr die reine und angewandte Mathematik (Crelles Journal) 01/2012; 2015(706). DOI:10.1515/crelle20130060 · 1.43 Impact Factor

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ABSTRACT: This note is an appendix to the paper âOsculating spaces and diophantine equations (with an Appendix by Pietro Corvaja and Umberto Zannier)â by M. Bolognesi and G. Pirola. Mathematische Nachrichten 09/2011; 284(13):1652  1657. DOI:10.1002/mana.200810300 · 0.68 Impact Factor

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ABSTRACT: We provide a lower bound for the number of distinct zeros of a sum 1 + u + v for two rational functions u, v, in term of the degree of u, v, which is sharp whenever u, v have few distinct zeros and poles compared to their degree. This sharpens the "abcdtheorem" of BrownawellMasser and Voloch in some cases which are sufficient to obtain new finiteness results on diophantine equations over function fields. For instance, we show that the Fermattype surface x a + y a + z c = 1 contains only finitely many rational or elliptic curves, provided a â„ 10 4 and c â„ 2; this provides special cases of a known conjecture of Bogomolov. Bulletin de la SocieÌteÌ matheÌmatique de France 01/2011; 139(4). · 0.42 Impact Factor

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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 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 · 1.29 Impact Factor

Source Available from: arxiv.org
Asian Journal of Mathematics 01/2010; 14(2010). DOI:10.4310/AJM.2010.v14.n4.a4 · 0.53 Impact Factor

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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 12/2009; 61(2009). DOI:10.2748/tmj/1264084501 · 0.32 Impact Factor

Source Available from: arxiv.org
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ABSTRACT: In 1988 ErdĂ¶s asked if the prime divisors of x
n
â 1 for all n = 1, 2, âŠ determine the given integer x; the problem was affirmatively answered by CorralesRodrigĂĄĂ±ez and Schoof (J Number Theory 64:276â290, 1997) [but a solution could also be deduced from an earlier result of Schinzel (Bull Acad Polon Sci 8:307â309, 2007)] together with its elliptic version. Analogously, Yamanoi (Forum Math 16:749â788, 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 â A essentially determines the pair (A, D). By making use of the main theorem of Noguchi (Forum Math 20:469â503, 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 â A
i
, i=1, 2, we classify the cases when the inclusion Suppf1*D1 Ă Suppf2* D2{{\rm{Supp}}\, f_1^*D_1\subset {\rm{Supp}}\, f_2^* D_2} holds. We shall remark in Â§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 149:431â451, 2002) to provide an arithmetic counterpart in the toric case. Mathematische Annalen 07/2009; 353(2):126. DOI:10.1007/s002080110692x · 1.13 Impact Factor

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ABSTRACT: We consider the question of existence of ramified covers over P1 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. MSC (2000): 57M12 (primary); 14H37, 11A41 (secondary). Elemente der Mathematik 11/2008; DOI:10.4171/EM/207

Developments in Mathematics 01/2008; 16. DOI:10.1007/9783211742808_8

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ABSTRACT: Let C be an irreducible algebraic curve in G2m; we are concerned with the maximal order m = m(C) of a torsion point on C. We suppose that C is defined over a number field k, that it is not a translate of an algebraic subgroup by a torsion point, and we denote by d its degree and by g its genus. It is known that <<k,Î”d2+Î” for any Î” > 0, which, as shown below, is nearly best possible if only the degree is taken into account. Here, by means of a new method, we prove an upper bound (actually for curves in G nm) which implies in particular m <<k,Î”k,Î” (d âd+g) 1+Î”. This renders the above result and for small g it improves on it. The appearance of the genus seems to be a new feature in this kind of problem. Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica E Applicazioni 01/2008; 19(1):7378. DOI:10.4171/RLM/509 · 0.52 Impact Factor

Source Available from: Carla Piazza
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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 formulĂŠinvolving both real and integer variables. While in the general case such formulĂŠ would be undecidable, the particular format of the formulĂŠ obtained in our translation allows combining decidability results stemming from both algebraic number theory and firstorder logic over (ĂŻÂŸÂż, 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

Source Available from: cs.nyu.edu
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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 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 formulĂŠ 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;

Source Available from: cs.nyu.edu
Hybrid Systems: Computation and Control, 10th International Workshop, HSCC 2007, Pisa, Italy, April 35, 2007, Proceedings; 01/2007

Source Available from: arxiv.org
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 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. Annali della Scuola normale superiore di Pisa, Classe di scienze 11/2006; 6(4). DOI:10.2422/20362145.2007.4.04 · 0.92 Impact Factor

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ABSTRACT: By means of a method that we recently introduced, we prove here some new results on integral points on surfaces, going sometimes beyond known results by Vojta. For instance, we show that if there are three numerically equivalent divisors at infinity which suitably intersect, then the integral points are not Zariskidense (known techniques, even in the particular case of P2, lead to the same conclusion only with four divisors at infinity). This will be derived as a special case of more general results and in turn will admit a concreteapplication to certain Diophantine equations. Such Diophantine equations include Sunit equations with variable coefficients, that is, linear equations of the forma(t)u+b(t)vCombining double low linec(t), where a(t), b(t), c(t) are suitable polynomials, to be solved in integer t and units u,v. They also include divisibility problems among values of certain polynomials at Sunit points. Finally, we will discuss the existence of certain rational curves on surfaces, related to the possible infinite families of integral points. International Mathematics Research Notices 01/2006; 2006:121. DOI:10.1155/IMRN/2006/98623 · 1.10 Impact Factor