On the equality of periods of Kontsevich-Zagier (Q2086407)

This paper is devoted to periods. \textit{M. Kontsevich} and \textit{D. Zagier} have given a concrete definition of periods as integrals [in: Mathematics unlimited -- 2001 and beyond. Berlin: Springer. 771--808 (2001; Zbl 1039.11002)]. They have conjectured that if a real period admits two integral representations, then one can pass from one to the other using only three operations: additions (by domains or integrands), change of variables and the Stokes formula. \textit{J. Viu-Sos} has proved [Int. J. Number Theory 17, No. 1, 147--174 (2021; Zbl 1467.14040)] that using these operations, any positive real period can be effectively written as the volume of a compact semi-algebraic set. Motivated by this result, the authors suggest in the present paper the following conjecture (and prove that it implies Kontsevich-Zagier's): \textbf{Conjecture.} Let \(K_1\) and \(K_2\) be compact top-dimensional semi-algebraic sets in \(\mathbb{R}^d\) with the same volume. Then one can transform \(K_1\) into \(K_2\) using only Cartesian product relations, semi-algebraic scissors congruences and algebraic volume-preserving maps respecting the Kontsevich-Zagier operations. The authors discuss also many related topics, including piecewise-linear geometry and a possible replacement of the Stokes formula.