Monomialization, Ramification and Regularity

研究者将继续他的工作在一元代数映射。研究者先前已经证明，通过执行映射序列，可以使光滑变体的一般有限映射成为光滑变体的局部单项式映射，这些映射序列是域和图像的光滑子变体的局部膨胀。然而，这些映射可能无法分离。研究者最近证明了从3折到表面的映射可以通过执行域和目标的光滑子变种的膨胀序列来局部单项式。所得到的品种是分离和适当的。在这个项目中，研究者试图将这个结果扩展到光滑变量之间的一般代数映射，并将之前的结果扩展到正特征，这在特征零中是有效的。研究者项目的目的是找到求解多项式方程组或解析方程的可计算方法。纯数学和应用数学中的许多重要问题都是这种形式。如果系统可以简化为一个单项式方程组，那么显式解就很容易给出。研究者发现了一种方法，通过执行非常简单和可计算的操作，可以将任何非退化系统局部化为单项式系统，从而可以很容易地读出系统的完整解。研究者将该方法推广到2个方程3个未知数的情况下，给出了可计算的全局解。研究者试图将这种方法扩展到一般大小的系统。

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.