Desingularization Algorithms: A Comparison from the Practical Point of View

Blanco, Rocio; Frühbis–Krüger, Anne

doi:10.1142/9789814383462_0002

Cited by 2 publications

References 16 publications

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A semi-canonical reduction for periods of Kontsevich–Zagier

Viu-Sos

2020

Int. J. Number Theory

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The [Formula: see text]-algebra of periods was introduced by Kontsevich and Zagier as complex numbers whose real and imaginary parts are values of absolutely convergent integrals of [Formula: see text]-rational functions over [Formula: see text]-semi-algebraic domains in [Formula: see text]. 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 Stokes formula. In this paper, we prove that every non-zero real period can be represented as the volume of a compact [Formula: see text]-semi-algebraic set obtained from any integral representation by an effective algorithm satisfying the rules allowed by the Kontsevich–Zagier period conjecture.

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A semi-canonical reduction for periods of Kontsevich–Zagier

Viu-Sos

2020

Int. J. Number Theory

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A semi-canonical reduction for periods of Kontsevich-Zagier

Viu-Sos

2015

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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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Desingularization Algorithms: A Comparison from the Practical Point of View

Cited by 2 publications

References 16 publications

A semi-canonical reduction for periods of Kontsevich–Zagier

A semi-canonical reduction for periods of Kontsevich–Zagier

A semi-canonical reduction for periods of Kontsevich-Zagier

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