Delta Matroids and the Geometry of Polynomials

Abstract

Dating back to Newton, mathematicians have long studied the interplay between the properties of polynomials as functions and their combinatorial properties. Recently, this has taken the form of stable and Lorentzian polynomials. In this thesis, we explore various facets of the connection between continuous and discrete properties of polynomials, with a particular focus on Delta matroids. This thesis is comprised of four projects, all related to Delta matroids or the geometry of polynomials. In the first project, we show that it is coNP-hard to decide whether a polynomial of fixed degree is real stable or log concave, while we can decide Lorentzianity in polynomial time.

Next, we turn our attention to the intersection of Lorentzianity and symmetric functions. Our main result is a reduction scheme that significantly reduces the complexity of testing for Lorentzianity. Using this method, we provide explicit semialgebraic descriptions of the spaces of Lorentzian symmetric polynomials and functions for degrees up to six. These techniques can also be applied to simplify the proofs to known cases of Lorentzian symmetric functions. We conclude by showing that some natural symmetric operators fail to preserve Lorentzianity which in turn highlights an inherent tension between symmetry in variables and the Lorentzian property. In the third chapter, we introduce valuated Delta matroids, a natural generalization of two objects of study in matroid theory: valuated matroids and Delta matroids. We show that these objects exhibit nice properties analogous to ordinary valuated matroids. We also show that these objects arise as the valuations of principal minors of a Hermitian matrix over a valued field, generalizing other forms of Delta matroid representability. Finally, we conclude with ongoing work trying to establish a theory analogous to Lorentzian polynomials for Delta matroids. We describe the motivations and applications that would come from such a result, and we discuss some partial progress towards it. This chapter aims to be a guide for any future research in the area, with emphasis placed on likely avenues for attacking the problem as well as warnings for potential pitfalls.