Integer Carath\'eodory results with bounded multiplicity

Abstract

The integer Carath\'eodory rank of a pointed rational cone $C$ is the smallest number $k$ such that every integer vector contained in $C$ is an integral non-negative combination of at most $k$ Hilbert basis elements. We investigate the integer Carath\'eodory rank of simplicial cones with respect to their multiplicity, i.e., the determinant of the integral generators of the cone. One of the main results states that simplicial cones with multiplicity bounded by five have the integral Carath\'eodory property, that is, the integer Carath\'eodory rank equals the dimension. Furthermore, we present a novel upper bound on the integer Carath\'eodory rank which depends on the dimension and the multiplicity. This bound improves upon the best known upper bound on the integer Carath\'eodory rank if the dimension exceeds the multiplicity. At last, we present special cones which have the integral Carath\'eodory property such as certain dual cones of Gorenstein cones.