Complexity in Commutative Algebra

研究者的主要目标是研究代数结构-环，理想甚至模块-因为它们经历平滑过程。这些转换使它们能够支持新的结构，包括分析结构。在代数的情况下，除数获得群结构，上同调趋于简化，这是奇异变异体化的重要步骤。在这些过程中有一种固有的兴趣，它将积分相关方程集合的解添加到结构中。找到这些方程，确定解集合的性质，并理解这些任务的复杂代价，是交换代数研究的中心领域。将多重性理论提供的数值控制（通常被视为对结构大小的分配）引入这种混合中，使得该领域与代数几何和计算代数的交互非常密集，这在技术上具有挑战性，但可能非常有益。研究者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。作为应用，研究者试图预测与平滑处理相关的精细技术在应用于解决几个感兴趣的问题时将如何执行，从而建议哪种方法组合提供更高的性能。它们也将被用来为这些问题中的一些导出普通的复杂性计数，而不需要已知的经典计数。交换代数主要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，特别是那些标记为科恩-麦考利型的结构。这些编码在其性质的推导/预测中具有令人难以置信的理论效率，并提供极好的计算经济性。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发现和分析它。这个建议集中在一个中心过程，即平滑变换。它将以Cohen-Macaulay案例为基础，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（甚至未知）算法的行为极限。这些发展中使用的方法和结果将用于本学科与代数几何、组合学、几何建模、数论和机器人技术的互动。

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.