A semi-canonical reduction for periods of Kontsevich-Zagier

Abstract

The $\overline{\mathbb{Q}}$-algebra of periods was introduced 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 ${\mathbb{R}}^d$. 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 Stokes formula. In this paper, we prove that every non-zero real period can be represented as the volume of a compact $\overline{\mathbb{Q}}\cap \mathbb{R}$-semi-algebraic set, obtained from any integral representation by an effective algorithm satisfying the rules allowed by the Kontsevich-Zagier period conjecture.

22 pages, 8 figures, 1 appendix with pseudo-codes describing the procedures. Major revision and correction of typos: the proof of the main result and associated procedures in sec. 2 are corrected and slightly improved; sec. 3 removed since its ideas are now part of the main procedure. To appear without the appendix in International Journal of Number Theory

References in corpus (1)

• Periods and elementary real numbers

Cited by in corpus (2)

• Computing the volume of compact semi-algebraic sets
• On the equality of periods of Kontsevich-Zagier