Complexity of Algebraic Structures

该项目介绍了一种从复杂性的角度研究某些代数结构的理论方面的方法。主要目标是研究代数结构--环、理想、甚至是模--在它们经历平滑过程时。这些转换使它们能够支持新的构造，包括解析构造。在代数的情况下，因子具有群结构，上同调趋于单调，这是奇异簇去单化的重要一步。其核心是对这些过程的内在兴趣，这些过程增加了结构，即积分依赖方程集合的解。寻找这些方程，确定解集合的性质，并了解这些任务的复杂性代价，是交换代数研究的中心领域。通过提出者和他的学生最近发现的多重性理论，将大小的度量分配给代数和结构本身是该项目的关键方面。作为应用，提出者试图预测当应用于几个感兴趣的问题的解决时，与平滑过程相关的精细技术将如何执行，从而建议哪种方法组合提供更高的性能。它们还将被用来推导出其中几个问题的普通复杂性计数，而这些问题之前没有已知的经典计数。交换代数，该提议的主题领域，首先是研究多项式方程组及其推广。它阐明了在这些系统中出现的几种结构，特别是那些被标记为Cohen-Macaulay类型的系统。这些编码在其属性的推导/预测中具有理论效率，并提供了极佳的计算经济性。通常，方程的完整自然集合在一开始是未知的，因此必须开发方法和过程来寻找和分析它。这一提议侧重于一个核心过程，即平滑转型。它将根据Cohen-Macaulay案例开发方法来预测闭包的性质，设计算法来找到它，并检查任意(甚至是未知的)算法的行为限制。开发的结果和方法将用于与代数几何、组合学、几何建模、数论和机器人学相结合的互动。

The project introduces an approach to the study of theoretical aspects of certain classes of algebraic structures from the perspective of complexity. The main objective is the study of algebraic structures--a ring, an ideal, or even a module--as they undergo smoothing processes. These transformations enable them to support new constructions, including analytic ones. In the case of algebras, divisors acquire a group structure, cohomology tends to slim down, and it is an essential step in the desingularization of singular varieties. At its core is the inherent interest in those processes that add to the structure the solutions of collections of equations of integral dependence. Finding these equations, determining the properties of the assemblage of solutions and understanding the complexity costs of these tasks is a central region of research for commutative algebra. The assignments of measures of size, via multiplicity theories recently discovered by the proposer and his students, to the algebras and to the construction itself are key aspects of the project. As applications, the proposer seeks to predict how delicate techniques associated to smoothing processes will perform when applied to the solution of several problems of interest, and thereby suggest which mix of methods offer higher performance. They will also be employed to derive ordinary complexity counts for several of these problems without previously known classical counts.Commutative algebra, the subject area of the proposal, is foremost the study of systems of polynomial equations, and of its generalizations. It has elucidated several structures that occur among such systems, particularly those tagged as of Cohen-Macaulay type. These encode theoretical efficiencies in the derivation/prediction of its properties and offer superb computational economies. Often the full natural set of equations is not known at the outset so that methods and processes must be developed to find and analyze it. This proposal is focused on one central process, that of smoothing transformation. It will develop methods, grounded on the Cohen-Macaulay case, to predict properties of the closure, devise algorithms to find it and examine the limits of the behavior of arbitrary (even unknown) algorithms. The results and methods developed will be used for interaction where the subject meets algebraic geometry, combinatorics, geometric modeling, number theory and robotics.