New Bounds for the Integer Carathéodory Rank

Abstract

Given a rational pointed $n$-dimensional cone $C$, we study the integer Carathéodory rank $\mathrm{CR}(C)$ and its asymptotic form $\mathrm{C}{\mathrm{R}}^{\mathrm{a}}(C)$, where we consider ``most'' integer vectors in the cone. The main result significantly improves the previously known upper bound for $\mathrm{C}{\mathrm{R}}^{\mathrm{a}}(C)$. We also study bounds on $\mathrm{CR}(C)$ in terms of $\Delta$, the maximal absolute $n\times n$ minor of the matrix given in an integral polyhedral representation of $C$. If $\Delta \in \{1,2\}$, we show $\mathrm{CR}(C)=n$, and prove upper bounds for simplicial cones, improving the best known upper bound on $\mathrm{CR}(C)$ for $\Delta \le n$.