On Finite Determinacy of Hypersurface Singularities and Matrices in Arbitrary Characteristic

Beschreibung

The thesis studies Mather-Yau theorem in positive characteristic and finite determinacy of matrices of power series over algebraically closed fields of arbitrary characteristic.
The thesis studies Mather-Yau theorem in positive characteristic and finite determinacy of hypersurface singularities and of matrices of power series over algebraically closed fields of arbitrary characteristic with respect to natural equivalence relations. It addresses the question that when such a matrix is determined by power series up to a certain order with respect to these equivalence relations, which is then called finitely determined. The first main result of the thesis is Mather-Yau theorem in positive characteristic. Over the complex numbers, the Mather-Yau theorem says that the contact class of an isolated hypersurface singularity is determined by its Tjurina algebra. It is known that this result is wrong over fields of positive characteristic. In this thesis we prove that over an algebraically closed field of arbitrary characteristic, the contact class of an isolated hypersurface singularity is determined by an associated Artinian algebra called the “higher Tjurina algebra”. A similar version is also stated for right equivalence. In addition, the thesis provides a sufficient condition for finite determinacy of matrices of power series over an algebraically closed field of arbitrary characteristic with respect to the action of the right group, i.e. the change of coordinates, and the actions by the multiplication by invertible matrices together with the change of coordinates. We also obtain explicit determinacy bounds for these group actions. Finally, we prove that for a complete intersection singularity I, isolatedness is equivalent to finite contact determinacy in arbitrary characteristic. We derive that 2τ(I) is then a contact determinacy bound, where τ(I) is the Tjurina number of I.

Autor*in

THUY HUONG PHAM
She obtained her PhD in Mathematics from the University of Kaiserslautern in 2016

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Mather-Yau theorem finite determinacy complete intersection singularities

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