Monomialization, Ramification and Regularity

研究人员将继续他在单项化代数映射方面的工作。研究者先前已经证明，一般的光滑簇的有限映射可以通过执行映射序列而成为光滑簇的局部单项映射，所述映射序列是域和像的光滑子簇的局部爆破。然而，这些映射可能无法分离。这位研究人员最近证明，通过执行结构域和目标的光滑亚种的放大序列，可以使从3折叠到表面的映射局部单项。由此产生的品种是分开的和适当的。在这个项目中，研究者试图将这一结果推广到光滑变元之间的一般代数映射，并将以前在特征零点有效的结果推广到正特征。研究者的项目目的是寻找多项式或解析方程组的可计算方法。纯数学和应用数学中的许多重要问题都是这种形式。如果系统可以归结为一个单项方程组，则可以很容易地给出显式解。研究人员已经找到了一种方法，通过执行非常简单和可计算的操作，将任何非退化系统局部化为单项系统，由此可以很容易地读出系统的完整解。研究人员将这种方法推广到3个未知数的2个方程的情况下，给出了可计算的整体解。研究人员试图将这种方法扩展到适用于一般规模的系统。

The investigator will continue his work on monomializing algebraic mappings. The investigator has previously shown that generically finite mappings of smooth varieties can be made to be locally monomial mappings of smooth varieties by performing sequences of mappings which are locally blowups of smooth subvarieties of the domain and image. These mappings may however fail to be separated. The investigator has more recently proven that mappings from a 3 fold to a surface can be made locally monomial by performing sequences of blowups of smooth subvarieties of the domain and target. The resulting varieties are separated and proper. In this project the investigator seeks to extend this result to general algebraic mappings between smooth varieties, and to extend the previous results, which are valid in characteristic zero, to positive characteristic.The purpose of the investigator's project is to find computible methods of solving systems of polynomial or analytic equations. Many important problems in pure and applied mathematics are of this form. If the system can be reduced to a system of monomial equations, explicit solutions can be given very easily. The investigator has found a method of reducing any nondegenerate system to a monomial system locally, by performing very simple and computible operations, from which a complete solution to the system can easily be read off. The investigator has extended this method to give a computable and global solution in the case of 2 equations in 3 unknowns. The investigator seeks to extend this method to work for systems of general size.