Complexity in Commutative Algebra

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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.