A semi-canonical reduction for periods of Kontsevich–Zagier

Abstract

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.

On the equality of periods of Kontsevich–Zagier

“…Also, we give some hints about the role that each of the three KZ-rules plays in the conjecture and related problems. The main ingredient is the following algorithmic result by the second author: [32,Thm. 1.1]).…”