Topics in Singularities, Valuation Theory and Commutative Algebra

University Of Missouri-Columbia, Columbia MO

Investigators

Abstract

Principal Investigator: Steven Cutkosky Proposal Number: DMS - 0240819 Institution: University of Missouri-Columbia Abstract: The investigator is studying the problem of monomalization of algebraic mappings. The conjecture is to prove that an algebraic mapping can be factored by a sequence of monoidal transforms, a locally monomial mapping, and a sequence of inverse monoidal transforms. This problem is being considered in both characteristic zero and characteristic p. The investigator has recently solved the local form of this conjecture in characteristic zero. The investigator is also working on applications to valuation theory. Another problem is the characterization of normal surface singularities with rational Poincare series. The Poincare series is associated to the exceptional divisors of a resolution of singularities. The investigator's work on monomialization will have important applications in other areas of mathematics and engineering. An immediate application is in the solution of systems of integral equations. One of the goals of the project is to find computable algorithms to transform systems of polynomial equations into monomial equations.