On the equality of periods of Kontsevich-Zagier. (English. French summary) Zbl 1510.11127
This paper is devoted to periods. M. Kontsevich and 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.
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):
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.
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):
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.
Reviewer: Stéphane Fischler (Paris)
MSC:
| 11J81 | Transcendence (general theory) |
| 14P10 | Semialgebraic sets and related spaces |
| 51M20 | Polyhedra and polytopes; regular figures, division of spaces |
| 51M25 | Length, area and volume in real or complex geometry |
| 52B45 | Dissections and valuations (Hilbert’s third problem, etc.) |
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