Monomialization, Ramification and Regularity

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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.