Higher order Jacobian matrix theory and invariants of singularities

Abstract

The higher order Nash blow-up of an algebraic variety replaces singular points with limits of certain spaces carrying higher order data associated to the variety at non-singular points. Inspired by the higher order Nash blow-up, we develop higher order Jacobian matrix theory, a generalization of the Jacobian matrix theory in the usual sense. The most significant contribution of this work is the higher order generalization of similarity transformation. It bridges the two long-standing, distinct research directions in the study of higher-order Jacobian matrices: the one from the higher order Nash blowup of a hypersurface, and the one representing a higher-order tangent map of a morphism. As an application, we use it to construct different contact and right invariants of isolated singularities, generalizing the Tjurina algebra and the Milnor algebra in singularity theory respectively. It is worth mentioning that the Tjurina algebra and the Milnor algebra were constructed using first-order derivatives, but now we are constructing invariants using arbitrary higher-order derivatives. The higher order Jacobian matrix theory also has potential applications and inspirations in multiple branches of mathematics: inspiring the establishment of higher order Hessian matrix theory and projective invariants, calculating explicit expression for the inverse of an automorphism of a power series ring, and performing explicit calculations in finite determinacy.