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ARITHMETIC GEOMETRY OF TORIC VARIETIES. METRICS, MEASURES AND HEIGHTS
, 2011
"... We show that the height of a toric variety with respect to a toric metrized line bundle can be expressed as the integral over a polytope of a certain adelic family of concave functions. To state and prove this result, we study the Arakelov geometry of toric varieties. In particular, we consider mo ..."
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We show that the height of a toric variety with respect to a toric metrized line bundle can be expressed as the integral over a polytope of a certain adelic family of concave functions. To state and prove this result, we study the Arakelov geometry of toric varieties. In particular, we consider models over a discrete valuation ring, metrized line bundles, and their associated measures and heights. We show that these notions can be translated in terms of convex analysis, and are closely related to objects like polyhedral complexes, concave functions, real MongeAmpère measures, and LegendreFenchel duality. We also present a closed formula for the integral over a polytope of a function of one variable composed with a linear form. This allows us to compute the height of toric varieties with respect to some interesting metrics arising from polytopes. We also compute the height of toric projective curves with respect to the FubiniStudy metric, and of some toric bundles.
ON THE ZEROS OF LINEAR RECURRENCE SEQUENCES
, 2009
"... Abstract. We improve by one exponential W. M. Schmidt’s estimate for the SkolemMahlerLech theorem on the number of arithmetic progressions describing the zeros of a linear recurrence sequence. 1. ..."
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Abstract. We improve by one exponential W. M. Schmidt’s estimate for the SkolemMahlerLech theorem on the number of arithmetic progressions describing the zeros of a linear recurrence sequence. 1.
Ultimate Positivity is Decidable for Simple Linear Recurrence Sequences?
"... Abstract. We consider the decidability and complexity of the Ultimate Positivity Problem, which asks whether all but finitely many terms of a given rational linear recurrence sequence (LRS) are positive. Using lower bounds in Diophantine approximation concerning sums of Sunits, we show that for sim ..."
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Abstract. We consider the decidability and complexity of the Ultimate Positivity Problem, which asks whether all but finitely many terms of a given rational linear recurrence sequence (LRS) are positive. Using lower bounds in Diophantine approximation concerning sums of Sunits, we show that for simple LRS (those whose characteristic polynomial has no repeated roots) the Ultimate Positivity Problem is decidable in polynomial space. If we restrict to simple LRS of a fixed order then we obtain a polynomialtime decision procedure. As a complexity lower bound we show that Ultimate Positivity for simple LRS is at least as hard as the decision problem for the universal theory of the reals: a problem that is known to lie between coNP and PSPACE. 1
SMALL POINTS ON RATIONAL SUBVARIETIES OF TORI.
, 2009
"... In this article we study the distribution of the small points of proper subvarieties of the torus Gn m defined over Q. For n = 1, the problem corresponds to finding lower bounds for the Weil height of an algebraic number. Let α be a nonzero algebraic number of degree D which is not a root of unity. ..."
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In this article we study the distribution of the small points of proper subvarieties of the torus Gn m defined over Q. For n = 1, the problem corresponds to finding lower bounds for the Weil height of an algebraic number. Let α be a nonzero algebraic number of degree D which is not a root of unity. Lehmer (see [Leh 1933])
ON A UNIFORM BOUND FOR THE NUMBER OF EXCEPTIONAL LINEAR SUBVARIETIES IN THE DYNAMICAL MORDELL–LANG CONJECTURE
"... Abstract. Let φ: P n → P n be a morphism of degree d ≥ 2 defined over C. The dynamical Mordell–Lang conjecture says that the intersection of an orbit Oφ(P) and a subvariety X ⊂ P n is usually finite. We consider the number of linear subvarieties L ⊂ P n such that the intersection Oφ(P) ∩ L is “larg ..."
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Abstract. Let φ: P n → P n be a morphism of degree d ≥ 2 defined over C. The dynamical Mordell–Lang conjecture says that the intersection of an orbit Oφ(P) and a subvariety X ⊂ P n is usually finite. We consider the number of linear subvarieties L ⊂ P n such that the intersection Oφ(P) ∩ L is “larger than expected. ” When φ is the d thpower map and the coordinates of P are multiplicatively independent, we prove that there are only finitely many linear subvarieties that are “superspanned ” by Oφ(P), and further that the number of such subvarieties is bounded by a function of n, independent of the point P and the degree d. More generally, we show that there exists a finite subset S, whose cardinality is bounded in terms of n, such that any n+1 points in Oφ(P)�S are in linear general position in P n. 1. The Dynamical Mordell–Lang Conjecture The classical Mordell conjecture says that a curve C of genus g ≥ 2 defined over a number field K has only finitely many Krational points. One may view C as embedded in its Jacobian J, and then Mordell’s conjecture may be reformulated as saying that C intersects the finitely generated group J(K) in only finitely many points. Taking this viewpoint, Lang
On the Positivity Problem for simple linear recurrence sequences
 In Proceedings of ICALP’14, 2014. CoRR, abs/1309.1550
"... Abstract. Given a linear recurrence sequence (LRS) over the integers, the Positivity Problem asks whether all terms of the sequence are positive. We show that, for simple LRS (those whose characteristic polynomial has no repeated roots) of order 9 or less, Positivity is decidable, with complexity ..."
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Abstract. Given a linear recurrence sequence (LRS) over the integers, the Positivity Problem asks whether all terms of the sequence are positive. We show that, for simple LRS (those whose characteristic polynomial has no repeated roots) of order 9 or less, Positivity is decidable, with complexity in the Counting Hierarchy. 1
ON THE REPRESENTATION OF FIBONACCI AND LUCAS NUMBERS IN AN INTEGER BASES
"... Abstract. Résumé. Nous présentons plusieurs théorèmes sur l’écriture des nombres entiers dans deux bases indépendantes. Nous dressons la liste complète des nombres de Fibonacci et des nombres de Lucas qui s’écrivent en binaire avec au plus quatre chiffres 1. Abstract. We discuss various results on t ..."
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Abstract. Résumé. Nous présentons plusieurs théorèmes sur l’écriture des nombres entiers dans deux bases indépendantes. Nous dressons la liste complète des nombres de Fibonacci et des nombres de Lucas qui s’écrivent en binaire avec au plus quatre chiffres 1. Abstract. We discuss various results on the representation of integers in two unrelated bases. We give the complete list of all the Fibonacci numbers and of all the Lucas numbers which have at most four digits 1 in their binary representation. 1.