The [Formula: see text]-algebra of periods was introduced by Kontsevich and Zagier as complex numbers whose real and imaginary parts are values of absolutely convergent integrals of [Formula: see text]-rational functions over [Formula: see text]-semi-algebraic domains in [Formula: see text]. 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 [Formula: see text]-semi-algebraic set obtained from any integral representation by an effective algorithm satisfying the rules allowed by the Kontsevich–Zagier period conjecture.

中文翻译：

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Kontsevich-Zagier 时期的半规范约简

[公式：见文本]-周期代数由 Kontsevich 和 Zagier 作为复数引入，其实部和虚部是 [公式：见文本]-有理函数在 [公式：见文本]-半上的绝对收敛积分的值- [公式：见正文]中的代数域。Kontsevich-Zagier 周期猜想确认给定周期的任何两个不同的积分表达式仅使用关于函数和域的合理性的三个规则通过有限的变换序列相关联：被积函数或域的积分相加，变量的变化和斯托克斯公式。在本文中，我们证明了每个非零实周期都可以表示为紧致的体积 [公式：