Algebra Commons^™

Simultaneous Zeros Of A System Of Two Quadratic Forms, Nandita Sahajpal

Theses and Dissertations--Mathematics

In this dissertation we investigate the existence of a nontrivial solution to a system of two quadratic forms over local fields and global fields. We specifically study a system of two quadratic forms over an arbitrary number field. The questions that are of particular interest are:

1. How many variables are necessary to guarantee a nontrivial zero to a system of two quadratic forms over a global field or a local field? In other words, what is the u-invariant of a pair of quadratic forms over any global or local field?
2. What is the relation between u-invariants of a …

Geometry Of Linear Subspace Arrangements With Connections To Matroid Theory, William Trok

Theses and Dissertations--Mathematics

This dissertation is devoted to the study of the geometric properties of subspace configurations, with an emphasis on configurations of points. One distinguishing feature is the widespread use of techniques from Matroid Theory and Combinatorial Optimization. In part we generalize a theorem of Edmond's about partitions of matroids in independent subsets. We then apply this to establish a conjectured bound on the Castelnuovo-Mumford regularity of a set of fat points.

We then study how the dimension of an ideal of point changes when intersected with a generic fat subspace. In particular we introduce the concept of a ``very unexpected hypersurface'' …

Algebraic And Geometric Properties Of Hierarchical Models, Aida Maraj

Theses and Dissertations--Mathematics

In this dissertation filtrations of ideals arising from hierarchical models in statistics related by a group action are are studied. These filtrations lead to ideals in polynomial rings in infinitely many variables, which require innovative tools. Regular languages and finite automata are used to prove and explicitly compute the rationality of some multivariate power series that record important quantitative information about the ideals. Some work regarding Markov bases for non-reducible models is shown, together with advances in the polyhedral geometry of binary hierarchical models.