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Convex Analysisの二,三の進展について

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

Optimal Transport: A Semi-Discrete Approach

The first goal of algebraic number theory is the generalization of the theorem on the unique representation of natural numbers as products of prime numbers to algebraic numbers. Gauss considered the ring \( \mathbb{Z}\left[ {\sqrt {{ - 1}} } \right] \) of all numbers of the form \( a + \sqrt {{ - 1b}} \) with \( a,\;b \in \mathbb{Z} \) and showed that \( \mathbb{Z}\left[ {\sqrt {{ - 1}} } \right] \) is a ring with unique factorization in prime elements (see §2.1). He introduced these numbers for the development of his theory of biquadratic residues. Another motivation for the study of the arithmetic of algebraic numbers comes from the theory of Tiophantine equations. For example, the quadratic form \( f({x_1},\;{x_2}) = x_1^2 - Dx_2^2 \) with \( D \in \mathbb{Z} \), \( \sqrt {{D\; \notin \;\mathbb{Z}}} \) can be written in the form \( \left( {{x_1} - \sqrt {{D{x_2}}} } \right)\left( {{x_1} + \sqrt {{D{x_2}}} } \right) \). Hence the question about the representation of integers by f(a 1, a 2) with \( {a_1},\;{a_2} \in \mathbb{Z} \) can be reformulated as a question of factorization of algebraic numbers of the form \( {a_1} + \sqrt {{D{a_2}}} \). These numbers form a module in the field \( \mathbb{Q}(\sqrt {D} ) \).

Basic Number Theory

Straight--Line Programs in Geometric Elimination Theory

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 Monge-Amp\`ere measures, and Legendre-Fenchel 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 Fubini-Study metric, and of some toric bundles.

Arithmetic Geometry of Toric Varieties: Metrics, Measures and Heights

/pdf/denominateurs-dans-le-theoreme-des-zeros-de-hilbert-5aq1dfwl73.pdf

Dénominateurs dans le théorème des zéros de Hilbert

Une approche méthodique pour la transcendance et l'indépendance algébrique de valeurs de fonctions analytiques☆

We continue with our study of the arithmetic geometry of toric varieties In this text, we study the positivity properties of metrized R-divisors in the toric setting For a toric metrized R-divisor, we give formulae for its arithmetic volume and its chi-arithmetic volume, and we characterize when it is arithmetically ample, nef, big or pseudo-effective, in terms of combinatorial data As an application, we prove a Dirichlet's unit theorem on toric varieties, we give a characterization for the existence of a Zariski decomposition of a toric metrized R-divisor, and we prove a toric arithmetic Fujita approximation theorem

Arithmetic positivity on toric varieties

In the frame of Mahler's method for algebraic independence we show that the algebraic relations over Q linking the values of functions solutions of a system of functional equations come from the algebraic relations between the functions themselves, by specialisation. We deduce some results on the linear independence of the q-regular functions.

Patrice Philippon

Papers

Arithmetic Geometry of Toric Varieties: Metrics, Measures and Heights

Dénominateurs dans le théorème des zéros de Hilbert

Une approche méthodique pour la transcendance et l'indépendance algébrique de valeurs de fonctions analytiques☆

Arithmetic positivity on toric varieties

Galois groups and automata