关于孔采维奇-扎吉尔的周期相等
On the equality of periods of Kontsevich-Zagier
摘要：
Kontsevich和Zagier将有效周期定义为复数，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周期猜想指出，任何两个不同的整式的周期是由一个有限序列的转换只使用三个规则尊重的合理性的功能和域：积分加法的被积或域，变量变换和斯托克斯公式。在本文中，我们介绍了两个几何解释这一猜想，被视为希尔伯特的第三个问题的推广，无论是紧凑的半代数集或有理多面体配备分段代数形式。基于已知的部分结果类似的希尔伯特第三问题，我们研究可能的几何方案来证明这个猜想和他们的潜在障碍。
英文摘要：
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.
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