Topics in Singularities, Valuation Theory and Commutative Algebra

主要研究人员：Steven Cutkosky建议编号：DMS-0240819机构：密苏里大学哥伦比亚分校摘要：研究人员正在研究代数映射的单值化问题。这个猜想是为了证明一个代数映射可以由一个么半群变换序列、一个局部单项映射和一个逆么半群变换序列分解。这个问题在特征零点和特征p点上都被考虑。研究者最近解决了这个猜想在特征零点的局部形式。这位研究人员还在研究估值理论的应用。另一个问题是用有理Poincare级数刻画法向曲面奇点。庞加莱级数与奇点分解的特殊因子有关。研究者在单项化方面的工作将在数学和工程的其他领域有重要的应用。一个直接的应用是求解积分方程组。该项目的目标之一是找到可计算的算法，将多项式方程系统转换为单项方程。

Principal Investigator: Steven Cutkosky Proposal Number: DMS - 0240819Institution: 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.