极大格自由体、测试集和Frobenius问题
Maximal lattice free bodies, test sets and the Frobenius problem
摘要：
极大格自由体是没有内部整点的极大多面体。斯卡夫开始研究最大晶格自由体相对于一个固定的矩阵的小平面法线。本文给出了计算整矩阵A的极大格自由体的一个有效算法。一个重要的组成部分是与A相关联的某个整数规划的测试集。该测试集可以使用代数方法来计算。作为应用，我们将三维Frobenius问题的Scarf-Shallcross算法推广到任意维数。在这种情况下，我们的方法的灵感来自新的算法由爱因斯坦，Lichtblau，Strzebonski和货车和Groebner基础的方法由Roune。
英文摘要：
Maximal lattice free bodies are maximal polytopes without interior integral points. Scarf initiated the study of maximal lattice free bodies relative to the facet normals in a fixed matrix. In this paper we give an efficient algorithm for computing the maximal lattice free bodies of an integral matrix A. An important ingredient is a test set for a certain integer program associated with A. This test set may be computed using algebraic methods. As an application we generalize the Scarf-Shallcross algorithm for the three-dimensional Frobenius problem to arbitrary dimension. In this context our method is inspired by the novel algorithm by Einstein, Lichtblau, Strzebonski and Wagon and the Groebner basis approach by Roune.
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