Complexity in Commutative Algebra

研究者的主要目标是研究代数结构--一个环，一个理想，甚至一个模--当它们经历平滑过程时。这些变换 使他们能够支持新的结构，包括分析的。在代数的情况下，因子获得了一个群结构，上同调趋于变细，这是奇异簇去奇异化的一个重要步骤。有一个内在的兴趣，在这些过程中，添加到结构的解决方案的集合方程integraldependence。寻找这些方程，确定解的集合的性质，并理解这些任务的复杂性代价，是交换代数研究的中心领域。 将多重性理论提供的数字控制引入这个混合物--广泛地被视为对结构大小的度量的分配--使一个技术上具有挑战性和潜在的非常有价值的活动恰好在该领域与代数几何和计算代数密切互动的地方。调查员介绍 从复杂性的角度研究某些代数结构的理论方面的方法。作为应用程序，调查人员试图预测如何微妙的技术与平滑过程将执行时，应用到几个感兴趣的问题的解决方案，从而建议哪些组合的方法提供更高的性能。他们也将被用来获得普通的复杂性计数的几个这些问题没有以前已知的经典counts.Commutative代数是最重要的研究系统的多项式方程，它的推广。它阐明了几个结构，发生在这样的系统，特别是那些标记为Cohen-Macaulay型。这些编码令人难以置信的理论效率在推导/预测其属性，并提供卓越的计算经济。通常一开始就不知道完整的自然方程组，因此必须开发方法和过程来找到和分析它。 它将发展的方法，基于科恩-麦考利的情况下，预测属性的“关闭”，设计算法来找到它，并检查的限制行为的任意[甚至未知] algorithm.The方法和结果中使用的这些发展将被用于互动的主题满足代数几何，组合数学，几何建模，数论和机器人。

The main objective of the investigator is the study ofalgebraic structures--a ring, an ideal or even a module--as theyundergo smoothing processes. These transformations enable them tosupport new constructions, including analytic ones. In the caseof algebras, divisors acquire a group structure, cohomology tends to slim down, and it is an essential step in the desingularization of singularvarieties. There is an inherent interest in those processes that add to thestructure the solutions of collections of equations of integraldependence. Finding these equations, determining theproperties of the assemblage of solutions and understanding thecomplexity costs of these tasks, is a central region of research for commutative algebra. Bringing into this mix the numerical controls provided by multiplicity theory--broadly seen as the assignment of measures of size to an structure--make for a technically challenging and potentially very rewarding activity smack right where the field interactsmostly intensively with algebraic geometry and computational algebra. The investigator introduces an approach to the study of theoretical aspects of certain classes of algebraic structures from the perspective of complexity. As applications, the investigator 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 severalof these problems without previously known classical counts.Commutative algebra is foremost the study of sytems of polynomial equations, and of its generalizations. It has elucidated severalstructures that occur among such systems, particularly those tagged asof Cohen-Macaulay type. These encode incredible theoretical efficiencies in thederivation/prediction of its properties and offer superb computationaleconomies. Often the full natural set of equations is not known atthe 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 ofthe `closure', devise algorithms to find it and examine the limits ofthe behavior of arbitrary [even unknown] algorithms.The methods and results used in these developments will be used forinteraction where the subject meets algebraic geometry, combinatorics, geometric modelling, number theory and robotics.