Kontsevich-Zagier周期的半正则约化
A semi-canonical reduction for periods of Kontsevich-Zagier
摘要：
周期${\overline{\mathbb Q}}$ -代数是由Kontsevich和Zagier作为复数引入的，其实部和虚部是${\mathbb R}^d$中${\mathbb Q}$ -半代数域上${\mathbb Q}$ -有理函数的绝对收敛积分的值。Kontsevich-Zagier周期猜想证明了给定周期的任意两个不同的整式通过一个有限的变换序列联系起来，而这一变换序列只需要使用三个关于函数和区域的合理性的规则：被积函数或区域的积分加法、变量的变量变换和Stokes公式。本文证明了每个非零实周期都可以表示为一个紧${\overline{\mathbb Q}}\cap{\mathbb R}$ -半代数集的体积，该集由任意整表示通过满足Kontsevich-Zagier周期猜想所允许的规则的有效算法得到.
英文摘要：
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.
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