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Entropy modulo a prime
Communications in Number Theory and Physics ( IF 1.9 ) Pub Date : 2021-06-01 , DOI: 10.4310/cntp.2021.v15.n2.a2
Tom Leinster 1
Affiliation  

Building on work of Kontsevich, we introduce a definition of the entropy of a finite probability distribution in which the ‘probabilities’ are integers modulo a prime $p$. The entropy, too, is an integer $\operatorname{mod} p$. Entropy $\operatorname{mod} p$ is shown to be uniquely characterized by a functional equation identical to the one that characterizes ordinary Shannon entropy. We also establish a sense in which certain real entropies have residues $\operatorname{mod} p$, connecting the concepts of entropy over $\mathbb{R}$ and over $\mathbb{Z} / p \mathbb{Z}$. Finally, entropy $\operatorname{mod} p$ is expressed as a polynomial which is shown to satisfy several identities, linking into work of Cathelineau, Elbaz–Vincent and Gangl on polylogarithms.

中文翻译:

熵模一个素数

基于 Kontsevich 的工作,我们引入了有限概率分布的熵的定义,其中“概率”是对素数 $p$ 取模的整数。熵也是一个整数 $\operatorname{mod} p$。熵 $\operatorname{mod} p$ 被证明是由一个与表征普通香农熵的函数方程相同的函数方程唯一表征的。我们还建立了一种意义,其中某些实熵具有残差 $\operatorname{mod} p$,连接了 $\mathbb{R}$ 和 $\mathbb{Z} / p \mathbb{Z}$ 上的熵概念. 最后,熵 $\operatorname{mod} p$ 表示为多项式,该多项式满足多个恒等式,链接到 Cathelineau、Elbaz-Vincent 和 Gangl 在多对数上的工作。
更新日期:2021-06-18
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