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Let $W$ be a finite group generated by reflections of a lattice $M$. If a lattice polytope $P \subset M \otimes_{\mathbb Z}\mathbb R$ is preserved by $W$, then we show that the quotient of the projective toric variety $X_P$ by $W$ is isomorphic to the toric variety $X_{P \cap D}$, where $D$ is a fundamental domain for the action of $W$. This answers a question of Horiguchi-Masuda-Shareshian-Song, and recovers results of Blume, of the second author, and of Gui-Hu-Liu., Comment: comments welcome!

The lattice of flats $\mathcal L_M$ of a matroid $M$ is combinatorially well-behaved and, when $M$ is realizable, admits a geometric model in the form of a "matroid Schubert variety". In contrast, the lattice of flats of a polymatroid exhibits many combinatorial pathologies and admits no similar geometric model. We address this situation by defining the lattice $\mathcal L_P$ of "combinatorial flats" of a polymatroid $P$. Combinatorially, $\mathcal L_P$ exhibits good behavior analogous to that of $\mathcal L_M$: it is graded, determines $P$ when $P$ is simple, and is top-heavy. When $P$ is realizable over a field of characteristic 0, we show that $\mathcal L_P$ is modelled by a "polymatroid Schubert variety". Our work generalizes a number of results of Ardila-Boocher and Huh-Wang on matroid Schubert varieties; however, the geometry of polymatroid Schubert varieties is noticeably more complicated than that of matroid Schubert varieties. Many natural questions remain open., Comment: 24 pages, 3 figures, comments welcome!

Schubert varieties of hyperplane arrangements, also known as matroid Schubert varieties, play an essential role in the proof of the Dowling-Wilson conjecture and in Kazhdan-Lusztig theory for matroids. We study these varieties as equivariant compactifications of affine spaces, and give necessary and sufficient conditions to characterize them. We also generalize the theory to include partial compactifications and morphisms between them. Our results resemble the correspondence between toric varieties and polyhedral fans., Comment: 23 pages; v2: Minor edits

We introduce the Bergman fan of a polymatroid and prove that the Chow ring of the Bergman fan is isomorphic to the Chow ring of the polymatroid. Using the Bergman fan, we establish the K\"ahler package for the Chow ring of the polymatroid, recovering and strengthening a result of Pagaria-Pezzoli., Comment: minor revisions

We demonstrate our implementation of a continuation method as described in \cite{HR2015} for solving polynomials systems. Given a sequence of (multi)homogeneous polynomials, the software "multiregeneration" outputs the respective (multi)degree in a wide range of cases and partial multidegree in all others. We use Python for the file processing, while Bertini is needed for the continuation. Moreover, parallelization options and several strategies for solving structured polynomial systems are available.

Speyer recognized that matroids encode the same data as a special class of tropical linear spaces and Shaw interpreted tropically certain basic matroid constructions; additionally, Frenk developed the perspective of tropical linear spaces as modules over an idempotent semifield. All together, this provides bridges between the combinatorics of matroids, the algebra of idempotent modules, and the geometry of tropical linear spaces. The goal of this paper is to strengthen and expand these bridges by systematically developing the idempotent module theory of matroids. Applications include a geometric interpretation of strong matroid maps and the factorization theorem; a generalized notion of strong matroid maps, via an embedding of the category of matroids into a category of module homomorphisms; a monotonicity property for the stable sum and stable intersection of tropical linear spaces; a novel perspective of fundamental transversal matroids; and a tropical analogue of reduced row echelon form., Comment: 22 pages; v3 minor corrections/clarifications; to appear in JPAA