On the equality of periods of Kontsevich-Zagier

TL;DR

This paper explores geometric interpretations of the Kontsevich-Zagier period conjecture, relating it to Hilbert's third problem, and discusses potential approaches and obstacles for proving the conjecture.

Contribution

It introduces two geometric frameworks for the period conjecture, extending the analogy with Hilbert's third problem, and analyzes possible proof strategies and challenges.

Findings

Proposes geometric interpretations involving semi-algebraic sets and rational polyhedra.

Connects the period conjecture to classical problems in geometry.

Identifies potential obstructions in proving the conjecture.

Abstract

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 $\mathbb{R}^d$. 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.