We consider a finite-dimensional vector space $W\subset K^E$ over a field K and a set E. We show that the set $\mathcal {C}(W)\subset 2^E$ of minimal supports of W are the circuits of a matroid on E. When the cardinality of K is large (compared to that of E), then the family of supports of W is a matroid. Afterwards we apply these results to tropical differential algebraic geometry (tdag), studying the set of supports of spaces of formal power series solutions $\text {Sol}(\Sigma )$ of systems of linear differential equations (ldes) $\Sigma$ in variables $x_1,\ldots ,x_n$ having coefficients in . If $\Sigma $ is of differential type zero, then the set $\mathcal {C}(Sol(\Sigma ))\subset (2^{\mathbb {N}^{m}})^n$ of minimal supports defines a matroid on $E=[n]\times \mathbb {N}^{m}$, and if the cardinality of K is large enough, then the set of supports is also a matroid on E. By applying the fundamental theorem of tdag (fttdag), we give a necessary condition under which the set of solutions $Sol(U)$ of a system U of tropical ldes is a matroid. We give a counterexample to the fttdag for systems $\Sigma $ of ldes over countable fields for which is not a matroid.

We generalize Baker–Bowler’s theory of matroids over tracts to orthogonal matroids, define orthogonal matroids with coefficients in tracts in terms of Wick functions, orthogonal signatures, circuit sets and orthogonal vector sets, and establish basic properties on functoriality, duality and minors. Our cryptomorphic definitions of orthogonal matroids over tracts provide proofs of several representation theorems for orthogonal matroids. In particular, we give a new proof that an orthogonal matroid is regular if and only if it is representable over ${\mathbb F}_2$ and ${\mathbb F}_3$, which was originally shown by Geelen [16], and we prove that an orthogonal matroid is representable over the sixth-root-of-unity partial field if and only if it is representable over ${\mathbb F}_3$ and ${\mathbb F}_4$.

This paper initiates the explicit study of face numbers of matroid polytopes and their computation. We prove that, for the large class of split matroid polytopes, their face numbers depend solely on the number of cyclic flats of each rank and size, together with information on the modular pairs of cyclic flats. We provide a formula which allows us to calculate $f$-vectors without the need of taking convex hulls or computing face lattices. We discuss the particular cases of sparse paving matroids and rank two matroids, which are of independent interest due to their appearances in other combinatorial and geometric settings.

We initiate the study of a class of polytopes, which we coin polypositroids, defined to be those polytopes that are simultaneously generalized permutohedra (or polymatroids) and alcoved polytopes. Whereas positroids are the matroids arising from the totally nonnegative Grassmannian, polypositroids are “positive” polymatroids. We parametrize polypositroids using Coxeter necklaces and balanced graphs, and describe the cone of polypositroids by extremal rays and facet inequalities. We introduce a notion of $(W,c)$-polypositroid for a finite Weyl group W and a choice of Coxeter element c. We connect the theory of $(W,c)$-polypositroids to cluster algebras of finite type and to generalized associahedra. We discuss membranes, which are certain triangulated 2-dimensional surfaces inside polypositroids. Membranes extend the notion of plabic graphs from positroids to polypositroids.

We prove that double Schubert polynomials have the saturated Newton polytope property. This settles a conjecture by Monical, Tokcan and Yong. Our ideas are motivated by the theory of multidegrees. We introduce a notion of standardization of ideals that enables us to study nonstandard multigradings. This allows us to show that the support of the multidegree polynomial of each Cohen–Macaulay prime ideal in a nonstandard multigrading, and in particular, that of each Schubert determinantal ideal is a discrete polymatroid.

We prove that projective spaces of Lorentzian and real stable polynomials are homeomorphic to Euclidean balls. This solves a conjecture of June Huh and the author. The proof utilises and refines a connection between the symmetric exclusion process in interacting particle systems and the geometry of polynomials.

We prove that the ranks of the subsets and the activities of the bases of a matroid define valuations for the subdivisions of a matroid polytope into smaller matroid polytopes.