Theta rank, levelness, and matroid minors
Francesco Grande, Raman Sanyal

TL;DR
This paper investigates the Theta rank and levelness of matroid base configurations, providing forbidden minor characterizations, especially for Theta-1 matroids, and exploring their algebraic and geometric properties.
Contribution
It offers a complete list of forbidden minors for Theta-1 matroids, generalizes series-parallel graphs, and links Theta rank to algebraic and geometric invariants.
Findings
Finite forbidden minor list for Theta-1 matroids
Characterization of Theta-1 matroids via algebraic invariants
Finite forbidden minors for k-level graphs and matroids
Abstract
The Theta rank of a finite point configuration is the maximal degree necessary for a sum-of-squares representation of a non-negative linear function on . This is an important invariant for polynomial optimization that is in general hard to determine. We study the Theta rank and levelness, a related discrete-geometric invariant, for matroid base configurations. It is shown that the class of matroids with bounded Theta rank or levelness is closed under taking minors. This allows for a characterization of matroids with bounded Theta rank or levelness in terms of forbidden minors. We give the complete (finite) list of excluded minors for Theta- matroids which generalizes the well-known series-parallel graphs. Moreover, the class of Theta- matroids can be characterized in terms of the degree of generation of the vanishing ideal and in terms of the psd rank for the associated…
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