  The Theta rank of a finite point configuration $V$ is the maximal degree
necessary for a sum-of-squares representation of a non-negative linear function
on $V$. 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-$1$ matroids which generalizes the
well-known series-parallel graphs. Moreover, the class of Theta-$1$ 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 matroid base polytope. We
further give a finite list of excluded minors for $k$-level graphs and matroids
and we investigate the graphs of Theta rank $2$.
