Degenerations of classical square matrices and their determinantal structure.

Abstract

In this thesis we study certain degenerations/specializations of the generic square matrix over a eld k of characteristic zero along its main related structures, such the determinant of the matrix, the ideal generated by its partial derivative, the polar map de ned by these derivatives, the Hessian matrix and the ideal of submaximal minors of the matrix. The degeneration types of the generic square matrix considered here are: (1) degeneration by \cloning" (repeating) a variable; (2) replacing a subset of entries by zeros, in a strategic layout; (3) further degeneration of the above types starting from certain specializations of the generic square matrix, such as the generic symmetric matrix and the generic square Hankel matrix. The focus in all these degenerations is in the invariants described above, highlighting on the homaloidal behavior of the matrix determinant. For this, we employ tools coming from commutative algebra, with emphasis on ideal theory and syzygy theory.