On the Algebraic and Geometric Multiplicity of Zero as a Hypergraph Eigenvalue

We consider the algebraic and geometric multiplicity of hypergraph eigenvalues, paying particular attention to nullities of hypertrees. Conjecture surrounds the relationship between these two multiplicites, but little work appears in the literature on this topic. Predominantly, we are interested in identifying the geometric structure of the nullvariety of certain hypergraph classes by listing the irreducible components and their accompanying dimensions. When applicable, we use this description to verify a conjecture relating the algebraic and geometric nullity. Furthermore, we refine these geometric structure tools to graph theoretic trees, defining a matroid on trees and relating skew zero forcing to the existence of nullvectors. Lastly, motivated by the algebraic nullity for general hyperdeterminants, we define “moment” and “unitary” ranks on numerical sequences, relating these notions to other well known properties of sequences