Stavros Garoufalidis
Abstract: There are two infinitesimal (i.e., additive) versions of the $K$-theory of a field $F$: one was introduced by Cathelineau, which is an $F$-module, and another one introduced by Bloch-Esnault, which is an $F^*$-module. Both versions are equipped with a regulator map, when $F$ is the field of complex numbers. In our short paper we will introduce an extended version of Cathelineau's group, and a complex-valued regulator map given by the entropy. We will also give a comparison map between our extended version and Cathelineau's group. Our results were motivated by two unrelated sources: Neumann's work on the extended Bloch group (which is isomorphic to indecomposable $K_3$ of the complex numbers), and the study of singularities of generating series of hypergeometric multisums.
Key words: infinitesimal K-theory, additive K-theory, regulators, entropy, 4-term relation, Stirling formula, binomial coefficients, infinitesimal polylogarithms, Bloch group, extended Bloch group.
Notes: 8 pages, 1 figures.