2,023 research outputs found

    Rational points on Grassmannians and unlikely intersections in tori

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    In this paper, we present an alternative proof of a finiteness theorem due to Bombieri, Masser and Zannier concerning intersections of a curve in the multiplicative group of dimension n with algebraic subgroups of dimension n-2. The proof uses a method introduced for the first time by Pila and Zannier to give an alternative proof of Manin-Mumford conjecture and a theorem to count points that satisfy a certain number of linear conditions with rational coefficients. This method has been largely used in many different problems in the context of "unlikely intersections".Comment: 16 page

    Constructing Permutation Rational Functions From Isogenies

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    A permutation rational function f∈Fq(x)f\in \mathbb{F}_q(x) is a rational function that induces a bijection on Fq\mathbb{F}_q, that is, for all y∈Fqy\in\mathbb{F}_q there exists exactly one x∈Fqx\in\mathbb{F}_q such that f(x)=yf(x)=y. Permutation rational functions are intimately related to exceptional rational functions, and more generally exceptional covers of the projective line, of which they form the first important example. In this paper, we show how to efficiently generate many permutation rational functions over large finite fields using isogenies of elliptic curves, and discuss some cryptographic applications. Our algorithm is based on Fried's modular interpretation of certain dihedral exceptional covers of the projective line (Cont. Math., 1994)

    On the Hilbert Property and the Fundamental Group of Algebraic Varieties

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    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

    Algebraic relations between solutions of Painlev\'e equations

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    We calculate model theoretic ranks of Painlev\'e equations in this article, showing in particular, that any equation in any of the Painlev\'e families has Morley rank one, extending results of Nagloo and Pillay (2011). We show that the type of the generic solution of any equation in the second Painlev\'e family is geometrically trivial, extending a result of Nagloo (2015). We also establish the orthogonality of various pairs of equations in the Painlev\'e families, showing at least generically, that all instances of nonorthogonality between equations in the same Painlev\'e family come from classically studied B{\"a}cklund transformations. For instance, we show that if at least one of α,β\alpha, \beta is transcendental, then PII(α)P_{II} (\alpha) is nonorthogonal to PII(β)P_{II} ( \beta ) if and only if α+β∈Z\alpha+ \beta \in \mathbb Z or α−β∈Z\alpha - \beta \in \mathbb Z. Our results have concrete interpretations in terms of characterizing the algebraic relations between solutions of Painlev\'e equations. We give similar results for orthogonality relations between equations in different Painlev\'e families, and formulate some general questions which extend conjectures of Nagloo and Pillay (2011) on transcendence and algebraic independence of solutions to Painlev\'e equations. We also apply our analysis of ranks to establish some orthogonality results for pairs of Painlev\'e equations from different families. For instance, we answer several open questions of Nagloo (2016), and in the process answer a question of Boalch (2012).Comment: This manuscript replaces and greatly expands a portion of arXiv:1608.0475

    Number Theory, Analysis and Geometry: In Memory of Serge Lang

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    Serge Lang was an iconic figure in mathematics, both for his own important work and for the indelible impact he left on the field of mathematics, on his students, and on his colleagues. Over the course of his career, Lang traversed a tremendous amount of mathematical ground. As he moved from subject to subject, he found analogies that led to important questions in such areas as number theory, arithmetic geometry and the theory of negatively curved spaces. Lang's conjectures will keep many mathematicians occupied far into the future. In the spirit of Lang’s vast contribution to mathematics, this memorial volume contains articles by prominent mathematicians in a variety of areas, namely number theory, analysis and geometry, representing Lang’s own breadth of interests. A special introduction by John Tate includes a brief and engaging account of Serge Lang’s life
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