Topics in Singularities, Valuation Theory and Commutative Algebra

主要研究者：Steven Cutkosky提案编号：DMS -0240819机构：密苏里-哥伦比亚大学摘要：研究者正在研究代数映射的单式化问题。猜想的目的是证明一个代数映射可以分解为一个么半群变换序列、一个局部么半群映射和一个逆么半群变换序列。这个问题正在考虑在两个特征零和特征p.调查人员最近解决了本地形式的这个猜想的特征零。该研究员还致力于估值理论的应用。另一个问题是用有理Poincare级数刻画法向曲面的奇异性。庞加莱级数与奇异点分解的例外因子有关。研究人员在单单性化方面的工作将在数学和工程的其他领域有重要的应用。直接应用是在积分方程组的解决方案。该项目的目标之一是找到可计算的算法，将多项式方程组转换为单项方程。

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.