New Bounds for the Integer Carathéodory Rank
研究了有理尖锥的整数Carathéodory秩及其渐近形式，显著改进了渐近秩的上界，并给出了基于最大子式Δ的秩的界。
Given a rational pointed n-dimensional cone C, we study the integer Caratheodory rank CR(C) and its asymptotic form CR^a(C), where we consider “most” integer vectors in the cone. The main result significantly improves the previously known upper bound for CR^a(C). We also study bounds on CR(C) in terms of ∆, the maximal absolute n × n minor of the matrix given in an integral polyhedral representation of C. If ∆ ∈ {1,2}, we show CR(C) = n, and prove upper bounds for simplicial cones, improving the best known upper bound on CR(C) for ∆ ≤ n.