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arXiv · 1903.06961

Entropy modulo a prime

Tom Leinster
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Abstract

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 mod p. Entropy 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 mod p, connecting the concepts of entropy over R and over Z/pZ. Finally, entropy 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.

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Tom Leinster. 2020-12-02. Entropy modulo a prime. https://arxiv.org/abs/1903.06961

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