On the equality of periods of Kontsevich-Zagier
arXiv:1912.01751
Abstract
Effective periods were defined by Kontsevich and Zagier as complex numbers whose real and imaginary parts are values of absolutely convergent integrals of -rational functions over -semi-algebraic domains in . 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.
17 pages, 1 figure. Improved exposition and corrected typos: Problems A and B are reformulated as conjectures and their relation with the KZ-conjecture is clarified; sec. 3 is extended; new references are added. To appear in Journal de Théorie des Nombres de Bordeaux