环面作用不变量理论中的多项式时间算法
Polynomial time algorithms in invariant theory for torus actions
摘要：
群在向量空间上的作用将向量空间划分为一组轨道。我们认为三个自然和有用的算法“同构”或“分类”的问题，即轨道平等，轨道封闭交叉，轨道封闭遏制。这些捕获并涉及数学，物理和计算机科学，优化和统计学中的各种问题。这些轨道问题扩展了更基本的零锥问题，其算法的复杂性在最近几年有了显着的进步。在本文中，我们开始研究这些问题的重点是交换群（即环面）的行动。我们解释了这种设置是如何从代数复杂性问题的动机，仍然是足够丰富的捕捉有趣的组合算法问题。虽然交换作用的结构理论很好地理解，但对于上述问题没有通用的有效算法。我们的主要结果是所有三个问题的多项式时间算法。我们还展示了如何有效地找到轨道的分离不变量，以及如何计算这些行动的生成有理不变量的系统（相比之下，对于多项式不变量，后者是已知的困难）。我们的技术是基于不变理论，线性规划和算法格理论的基本结果的组合。
英文摘要：
An action of a group on a vector space partitions the latter into a set of orbits. We consider three natural and useful algorithmic "isomorphism" or "classification" problems, namely, orbit equality, orbit closure intersection, and orbit closure containment. These capture and relate to a variety of problems within mathematics, physics and computer science, optimization and statistics. These orbit problems extend the more basic null cone problem, whose algorithmic complexity has seen significant progress in recent years. In this paper, we initiate a study of these problems by focusing on the actions of commutative groups (namely, tori). We explain how this setting is motivated from questions in algebraic complexity, and is still rich enough to capture interesting combinatorial algorithmic problems. While the structural theory of commutative actions is well understood, no general efficient algorithms were known for the aforementioned problems. Our main results are polynomial time algorithms for all three problems. We also show how to efficiently find separating invariants for orbits, and how to compute systems of generating rational invariants for these actions (in contrast, for polynomial invariants the latter is known to be hard). Our techniques are based on a combination of fundamental results in invariant theory, linear programming, and algorithmic lattice theory.
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