On the equality of periods of Kontsevich–Zagier

Cresson, Jacky; Viu-Sos, Juan

Journal de théorie des nombres de Bordeaux, Tome 34 (2022) no. 2, pp. 323-343.

Les périodes effectives furent définies par Kontsevich et Zagier comme étant les nombres complexes dont les parties réelle et imaginaire sont valeurs d’intégrales absolument convergentes de fonctions $\mathbb{Q}$-rationnelles sur des domaines $\mathbb{Q}$-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 introduisons deux interprétations géométriques de cette conjecture, vue comme une généralisation du 3ème problème de Hilbert soit pour des ensembles semi-algébriques compacts soit pour des polyèdres rationnels munis d’une forme volume algébrique par morceaux. Basés sur des résultats partiels connus pour des problèmes de Hilbert analogues, nous étudions des schémas géométriques possibles pour obtenir une preuve de la conjecture et ses obstructions potentielles.

Effective periods were defined by Kontsevich and Zagier as complex numbers whose real and imaginary parts are values of absolutely convergent integrals of $\mathbb{Q}$-rational functions over $\mathbb{Q}$-semi-algebraic domains in ${ℝ}^d$. The Kontsevich–Zagier period conjecture states that any two different integral expressions of a period are related by a finite sequence of transformations only using three rules respecting the rationality of functions and domains: integral addition by integrands or domains, change of variables and Stokes’ formula.

In this paper, we introduce two geometric interpretations of this conjecture, seen as a generalization of Hilbert’s third problem involving either compact semi-algebraic sets or rational polyhedra equipped with piece-wise algebraic forms. Based on known partial results for analogous Hilbert’s third problems, we study possible geometric schemes to prove this conjecture and their potential obstructions.

Classification : 11J81, 51M25, 52B45, 14P10
Mots clés : Periods, Kontsevich–Zagier period conjecture, Hilbert third problem.

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