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

Integer Carath\'eodory results with bounded multiplicity