On the effective algebraic geometry of determinantal varieties

Abstract

The central objects of study in this thesis are varieties defined by rank deficiencies of matrices with polynomial entries, which are known as determinantal varieties. We consider effective questions about these varieties from two distinct perspectives. The first part of this thesis concerns computational results in the arithmetic complexity model, where basic field operations have unit cost. From classical results on free resolutions of determinantal ideals, we extract information about the syzygy modules of determinantal ideals, and use this information to adapt existing general-purpose Gröbner basis algorithms to determinantal ideals. This leads us to new algorithms to solve zero-dimensional determinantal polynomial systems and polynomial optimisation problems. We give complexity analyses of our algorithms, and in so doing, show that they improve upon the state-of-the-art. The central problem considered in the second part of this thesis is to bound heights of determinantal varieties. By way of new constructions in arithmetic integral geometry, we give exact values for Mahler measures of determinantal resultants. Coupled with a determinantal analogue of the standard u-resultant, this enables us to bound heights of Chow forms of determinantal varieties.