论文标题
整数Carathéodory等级的新界限
New Bounds for the Integer Carathéodory Rank
论文作者
论文摘要
鉴于有理尖的$ n $维锥$ c $，我们研究了整数Carathéodory等级$ \ permatatorName {cr}（c）$及其渐近形式$ \ operatatorName {cr^{cr^{\ rm a}}（c）（c）$主要结果显着改善了以前已知的上限，以$ \ operatatorName {cr^{\ rm a}}}（c）$。我们还以$δ$为$ \ operatatorName {cr}（c）$在$δ$中研究界限，这是$ c $的整体多面体表示中给出的最大绝对$ n \ times n $ n $。如果$δ\ in \ lbrace 1,2 \ rbrace $，我们显示$ \ operatatorName {cr}（c）= n $，并证明了上限的简单锥，改善了$ \ operatatorNamame {cr}（c）（c）$ $Δ\ leq leq n $上的$ \ operatatorNAME {cr}（c）$。
Given a rational pointed $n$-dimensional cone $C$, we study the integer Carathéodory rank $\operatorname{CR}(C)$ and its asymptotic form $\operatorname{CR^{\rm a}}(C)$, where we consider ``most'' integer vectors in the cone. The main result significantly improves the previously known upper bound for $\operatorname{CR^{\rm a}}(C)$. We also study bounds on $\operatorname{CR}(C)$ in terms of $Δ$, the maximal absolute $n\times n$ minor of the matrix given in an integral polyhedral representation of $C$. If $Δ\in\lbrace 1,2\rbrace$, we show $\operatorname{CR}(C) = n$, and prove upper bounds for simplicial cones, improving the best known upper bound on $\operatorname{CR}(C)$ for $Δ\leq n$.
