Periods and elementary real numbers

Yoshinaga, Masahiko

doi:10.48550/arxiv.0805.0349

Cited by 5 publications

(5 citation statements)

References 9 publications

(16 reference statements)

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“…We can extend the Theorem 1.1 for the whole set of periods P kz ⊂ C considering representations of the real and imaginary part respectively. Such a representation for a period p is called a geometric semi-canonical representation of p. This kind of representation was also suggested by M. Yoshinaga in [Yos08,p. 13] and assumed by J. Wan in [Wan11] in order to develop a degree theory for periods.…”

Section: Introductionmentioning

confidence: 91%

A semi-canonical reduction for periods of Kontsevich-Zagier

Viu-Sos

2015

Preprint

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The Q-algebra of periods was introduced by Kontsevich and Zagier [KZ01] as complex numbers whose real and imaginary parts are values of absolutely convergent integrals of Q-rational functions over Q-semi-algebraic domains in R d . The Kontsevich-Zagier period conjecture affirms that any two different integral expressions of a given period are related by a finite sequence of transformations only using three rules respecting the rationality of the functions and domains: additions of integrals by integrands or domains, change of variables and Stoke's formula.In this paper, we prove that every non-zero real period can be represented as the volume of a compact Q ∩ R-semi-algebraic set, obtained from any integral representation by an effective algorithm satisfying the rules allowed by the Kontsevich-Zagier period conjecture.Résumé. La Q-algèbre des périodes fut introduite par Kontsevich et Zagier [KZ01] comme les nombres complexes dont les parties réelle et imaginaire sont valeurs d'intégrales absolument convergentes de fonctions Q-rationnelles sur des domaines Q-semi-algébriques dans R d . La conjecture des périodes de Kontsevich-Zagier affirme que si une période admet deux représentations intégrales, alors elles sont reliées par une suite finie d'opérations en utilisant uniquement trois règles respectant la rationalité des fonctions et domaines : sommes d'intégrales par intégrandes ou domaines, changement de variables et formule de Stokes.Dans cet article, nous démontrons que toute période réelle non nulle peut être représentée comme le volume d'un ensemble Q ∩ R-semi-algébrique compact, obtenu à partir de n'importe quelle représentation intégrale via un algorithme effectif en respectant les règles permises par la conjecture des périodes de Kontsevich-Zagier.

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Section: Introductionmentioning

confidence: 91%

A semi-canonical reduction for periods of Kontsevich-Zagier

Viu-Sos

2015

Preprint

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“…On the other hand, it is conjectured that the basis of the natural logarithm e and Euler's constant γ E are not periods. Although there are uncountably many numbers, which are not periods, only very recently an example for a number which is not a period has been found [18]. We need a few basic properties of periods: The set of periods P is a Q-algebra [17,19].…”

Section: Periodsmentioning

confidence: 99%

Blowing up Feynman integrals

Bogner

Weinzierl

2008

Nuclear Physics B - Proceedings Supplements

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In this talk we discuss sector decomposition. This is a method to disentangle overlapping singularities through a sequence of blow-ups. We report on an open-source implementation of this algorithm to compute numerically the Laurent expansion of divergent multi-loop integrals. We also show how this method can be used to prove a theorem which relates the coefficients of the Laurent series of dimensionally regulated multi-loop integrals to periods.Comment: 6 pages, talk given at the conference "Loops and Legs in Quantum Field Theory", Sondershausen, April 200

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Section: Feynman Integrals and Periodsmentioning

confidence: 99%

Introduction to Feynman integrals

Weinzierl¹

2013

Geometric and Topological Methods for Quantum Field Theory

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In these lectures I will give an introduction to Feynman integrals. In the first part of the course I review the basics of the perturbative expansion in quantum field theories. In the second part of the course I will discuss more advanced topics: Mathematical aspects of loop integrals related to periods, shuffle algebras and multiple polylogarithms are covered as well as practical algorithms for evaluating Feynman integrals.

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Periods and elementary real numbers

Cited by 5 publications

References 9 publications

A semi-canonical reduction for periods of Kontsevich-Zagier

A semi-canonical reduction for periods of Kontsevich-Zagier

Blowing up Feynman integrals

Introduction to Feynman integrals

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