A complete classification of unimodular valuations on the set of lattice polygons with values in the spaces of polynomials and formal power series, respectively, is established. The valuations are classified in terms of their behavior with respect to dilation using extensions to unbounded polyhedra and basic invariant theory.

We prove that the initial degenerations of the flag variety admit closed immersions into finite inverse limits of flag matroid strata, where the diagrams are derived from matroidal subdivisions of a suitable flag matroid polytope. As an application, we prove that the initial degenerations of $\mathrm{F}\ell^{\circ}(n)$–the open subvariety of the complete flag variety $\mathrm{F}\ell(n)$ consisting of flags in general position—are smooth and irreducible when $n\leq 4$. We also study the Chow quotient of $\mathrm{F}\ell(n)$ by the diagonal torus of $\textrm{PGL}(n)$ and show that, for $n=4$, this is a log crepant resolution of its log canonical model.

We study the hypersimplex under the action of the symmetric group $S_n$ by coordinate permutation. We prove that its equivariant volume, given by the evaluation of its equivariant $H^*$-series at $1$, is the permutation character of decorated ordered set partitions under the natural action of $S_n$. This verifies a conjecture of Stapledon for the hypersimplex. To prove this result, we give a formula for the coefficients of the $H^*$-polynomial. Additionally, for the $(2,n)$-hypersimplex, we use this formula to show that trivial character need not appear as a direct summand of a coefficient of the $H^*$-polynomial, which gives a family of counterexamples to a different conjecture of Stapledon.

Building on the correspondence between finitely axiomatised theories in Łukasiewicz logic and rational polyhedra, we prove that the unification type of the fragment of Łukasiewicz logic with $n\geqslant 2$ variables is nullary. This solves a problem left open by V. Marra and L. Spada [Ann. Pure Appl. Logic 164 (2013), pp. 192–210]. Furthermore, we refine the study of unification with bounds on the number of variables. Our proposal distinguishes the number m of variables allowed in the problem and the number n in the solution. We prove that the unification type of Łukasiewicz logic for all $m,n \geqslant 2$ is nullary.

An empty simplex is a lattice simplex in which vertices are the only lattice points. We show two constructions leading to the first known empty simplices of width larger than their dimension:

• ◦ We introduce cyclotomic simplices and exhaustively compute all the cyclotomic simplices of dimension $10$ and volume up to $2^{31}$. Among them, we find five empty ones of width $11$ and none of larger width.

• ◦ Using circulant matrices of a very specific form, we construct empty simplices of arbitrary dimension d and width growing asymptotically as $d/\operatorname {\mathrm {arcsinh}}(1) \sim 1.1346\,d$.

We conduct a systematic study of the Ehrhart theory of certain slices of rectangular prisms. Our polytopes are generalizations of the hypersimplex and are contained in the larger class of polypositroids introduced by Lam and Postnikov; moreover, they coincide with polymatroids satisfying the strong exchange property up to an affinity. We give a combinatorial formula for all the Ehrhart coefficients in terms of the number of weighted permutations satisfying certain compatibility properties. This result proves that all these polytopes are Ehrhart positive. Additionally, via an extension of a result by Early and Kim, we give a combinatorial interpretation for all the coefficients of the $h^*$-polynomial. All of our results provide a combinatorial understanding of the Hilbert functions and the h-vectors of all algebras of Veronese type, a problem that had remained elusive up to this point. A variety of applications are discussed, including expressions for the volumes of these slices of prisms as weighted combinations of Eulerian numbers; some extensions of Laplace’s result on the combinatorial interpretation of the volume of the hypersimplex; a multivariate generalization of the flag Eulerian numbers and refinements; and a short proof of the Ehrhart positivity of the independence polytope of all uniform matroids.

The most fundamental example of mirror symmetry compares the Fermat hypersurfaces in $\mathbb {P}^n$ and $\mathbb {P}^n/G$, where G is a finite group that acts on $\mathbb {P}^n$ and preserves the Fermat hypersurface. We generalize this to hypersurfaces in Grassmannians, where the picture is richer and more complex. There is a finite group G that acts on the Grassmannian $\operatorname {{\mathrm {Gr}}}(n,r)$ and preserves an appropriate Calabi–Yau hypersurface. We establish how mirror symmetry, toric degenerations, blow-ups and variation of GIT relate the Calabi–Yau hypersurfaces inside $\operatorname {{\mathrm {Gr}}}(n,r)$ and $\operatorname {{\mathrm {Gr}}}(n,r)/G$. This allows us to describe a compactification of the Eguchi–Hori–Xiong mirror to the Grassmannian, inside a blow-up of the quotient of the Grassmannian by G.

We give an upper bound on the volume $\operatorname {vol}(P^*)$ of a polytope $P^*$ dual to a d-dimensional lattice polytope P with exactly one interior lattice point in each dimension d. This bound, expressed in terms of the Sylvester sequence, is sharp and achieved by the dual to a particular reflexive simplex. Our result implies a sharp upper bound on the volume of a d-dimensional reflexive polytope. Translated into toric geometry, this gives a sharp upper bound on the anti-canonical degree $(-K_X)^d$ of a d-dimensional Fano toric variety X with at worst canonical singularities.

In 2006, Kenyon and Okounkov Kenyon and Okounkov [12] computed the moduli space of Harnack curves of degree d in ${\mathbb {C}\mathbb {P}}^2$. We generalise their construction to any projective toric surface and show that the moduli space ${\mathcal {H}_\Delta }$ of Harnack curves with Newton polygon $\Delta $ is diffeomorphic to ${\mathbb {R}}^{m-3}\times {\mathbb {R}}_{\geq 0}^{n+g-m}$, where $\Delta $ has m edges, g interior lattice points and n boundary lattice points. This solves a conjecture of Crétois and Lang. The main result uses abstract tropical curves to construct a compactification of this moduli space where additional points correspond to collections of curves that can be patchworked together to produce a curve in ${\mathcal {H}_\Delta }$. This compactification has a natural stratification with the same poset as the secondary polytope of $\Delta $.

Let $M$ be a regular matroid. The Jacobian group $\text{Jac}(M)$ of $M$ is a finite abelian group whose cardinality is equal to the number of bases of $M$. This group generalizes the definition of the Jacobian group (also known as the critical group or sandpile group) $\operatorname{Jac}(G)$ of a graph $G$ (in which case bases of the corresponding regular matroid are spanning trees of $G$). There are many explicit combinatorial bijections in the literature between the Jacobian group of a graph $\text{Jac}(G)$ and spanning trees. However, most of the known bijections use vertices of $G$ in some essential way and are inherently ‘nonmatroidal’. In this paper, we construct a family of explicit and easy-to-describe bijections between the Jacobian group of a regular matroid $M$ and bases of $M$, many instances of which are new even in the case of graphs. We first describe our family of bijections in a purely combinatorial way in terms of orientations; more specifically, we prove that the Jacobian group of $M$ admits a canonical simply transitive action on the set ${\mathcal{G}}(M)$ of circuit–cocircuit reversal classes of $M$, and then define a family of combinatorial bijections $\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70E}^{\ast }}$ between ${\mathcal{G}}(M)$ and bases of $M$. (Here $\unicode[STIX]{x1D70E}$ (respectively $\unicode[STIX]{x1D70E}^{\ast }$) is an acyclic signature of the set of circuits (respectively cocircuits) of $M$.) We then give a geometric interpretation of each such map $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70E}^{\ast }}$ in terms of zonotopal subdivisions which is used to verify that $\unicode[STIX]{x1D6FD}$ is indeed a bijection. Finally, we give a combinatorial interpretation of lattice points in the zonotope $Z$; by passing to dilations we obtain a new derivation of Stanley’s formula linking the Ehrhart polynomial of $Z$ to the Tutte polynomial of $M$.

Fixing a positive integer r and $0 \les k \les r-1$, define $f^{\langle r,k \rangle }$ for every formal power series f as $ f(x) = f^{\langle r,0 \rangle }(x^r)+xf^{\langle r,1 \rangle }(x^r)+ \cdots +x^{r-1}f^{\langle r,r-1 \rangle }(x^r).$ Jochemko recently showed that the polynomial $U^{n}_{r,k}\, h(x) := ( (1+x+\cdots +x^{r-1})^{n} h(x) )^{\langle r,k \rangle }$ has only non-positive zeros for any $r \ges \deg h(x) -k$ and any positive integer n. As a consequence, Jochemko confirmed a conjecture of Beck and Stapledon on the Ehrhart polynomial $h(x)$ of a lattice polytope of dimension n, which states that $U^{n}_{r,0}\,h(x)$ has only negative, real zeros whenever $r\ges n$. In this paper, we provide an alternative approach to Beck and Stapledon's conjecture by proving the following general result: if the polynomial sequence $( h^{\langle r,r-i \rangle }(x))_{1\les i \les r}$ is interlacing, so is $( U^{n}_{r,r-i}\, h(x) )_{1\les i \les r}$. Our result has many other interesting applications. In particular, this enables us to give a new proof of Savage and Visontai's result on the interlacing property of some refinements of the descent generating functions for coloured permutations. Besides, we derive a Carlitz identity for refined coloured permutations.

A result of Haglund implies that the $(q,t)$-bigraded Hilbert series of the space of diagonal harmonics is a $(q,t)$-Ehrhart function of the flow polytope of a complete graph with netflow vector $(-n,1,\ldots ,1)$. We study the $(q,t)$-Ehrhart functions of flow polytopes of threshold graphs with arbitrary netflow vectors. Our results generalize previously known specializations of the mentioned bigraded Hilbert series at $t=1$, $0$, and $q^{-1}$. As a corollary to our results, we obtain a proof of a conjecture of Armstrong, Garsia, Haglund, Rhoades, and Sagan about the $(q,q^{-1})$-Ehrhart function of the flow polytope of a complete graph with an arbitrary netflow vector.

We prove that the monodromy group of a reduced irreducible square system of general polynomial equations equals the symmetric group. This is a natural first step towards the Galois theory of general systems of polynomial equations, because arbitrary systems split into reduced irreducible ones upon monomial changes of variables. In particular, our result proves the multivariate version of the Abel–Ruffini theorem: the classification of general systems of equations solvable by radicals reduces to the classification of lattice polytopes of mixed volume 4 (which we prove to be finite in every dimension). We also notice that the monodromy of every general system of equations is either symmetric or imprimitive. The proof is based on a new result of independent importance regarding dual defectiveness of systems of equations: the discriminant of a reduced irreducible square system of general polynomial equations is a hypersurface unless the system is linear up to a monomial change of variables.

In 2012, Gubeladze (Adv. Math. 2012) introduced the notion of $k$ -convex-normal polytopes to show that integral polytopes all of whose edges are longer than $4d(d+1)$ have the integer decomposition property. In the first part of this paper we show that for lattice polytopes there is no diòerence between $k$ - and $(k+1)$ -convex-normality (for $k\ge 3$ ) and improve the bound to $2d(d+1)$ . In the second part we extend the definition to pairs of polytopes. Given two rational polytopes $P$ and $\text{Q}$ , where the normal fan of $P$ is a refinement of the normal fan of $\text{Q}$ , if every edge ${{e}_{P}}$ of $P$ is at least $d$ times as long as the corresponding face (edge or vertex) ${{e}_{\text{Q}}}$ of $\text{Q}$ , then $(P+\text{Q})\cap {{\mathbb{Z}}^{d}}=(P\cap {{\mathbb{Z}}^{d}})+(\text{Q}\cap {{\mathbb{Z}}^{d}})$ .

Toric quiver varieties (moduli spaces of quiver representations) are studied. Given a quiver and a weight, there is an associated quasi-projective toric variety together with a canonical embedding into projective space. It is shown that for a quiver with no oriented cycles the homogeneous ideal of this embedded projective variety is generated by elements of degree at most 3. In each fixed dimension d up to isomorphism there are only finitely many d-dimensional toric quiver varieties. A procedure for their classification is outlined.

A rational polyhedron$P\subseteq {\mathbb{R^n}}$ is a finite union of simplexes in ${\mathbb{R^n}}$ with rational vertices. P is said to be $\mathbb Z$-homeomorphic to the rational polyhedron $Q\subseteq {\mathbb{R^{\it m}}}$ if there is a piecewise linear homeomorphism η of P onto Q such that each linear piece of η and η−1 has integer coefficients. When n=m, $\mathbb Z$-homeomorphism amounts to continuous $\mathcal{G}_n$-equidissectability, where $\mathcal{G}_n=GL(n,\mathbb Z) \ltimes \mathbb Z^{n}$ is the affine group over the integers, i.e., the group of all affinities on $\mathbb{R^{n}}$ that leave the lattice $\mathbb Z^{n}$ invariant. $\mathcal{G}_n$ yields a geometry on the set of rational polyhedra. For each d=0,1,2,. . ., we define a rational measure λd on the set of rational polyhedra, and show that any two $\mathbb Z$-homeomorphic rational polyhedra $$P\subseteq {\mathbb{R^n}}$$ and $Q\subseteq {\mathbb{R^{\it m}}}$ satisfy $\lambda_d(P)=\lambda_d(Q)$. $\lambda_n(P)$ coincides with the n-dimensional Lebesgue measure of P. If 0 ≤ dim P=d < n then λd(P)>0. For rational d-simplexes T lying in the same d-dimensional affine subspace of ${\mathbb{R^{\it n}}, $\lambda_d(T)$$ is proportional to the d-dimensional Hausdorff measure of T. We characterize λd among all unimodular invariant valuations.

The concept of centrally symmetric configurations of integer matrices is introduced. We study the problem when the toric ring of a centrally symmetric configuration is normal and when it is Gorenstein. In addition, Gröbner bases of toric ideals of centrally symmetric configurations are discussed. Special attention is given to centrally symmetric configurations of unimodular matrices and to those of incidence matrices of finite graphs.

Let k be an algebraically closed field of characteristic zero. I. M. Musson and M. Van den Bergh (Mem. Amer. Math. Soc., vol. 136, 1998, p. 650) classify primitive ideals for rings of torus invariant differential operators. This classification applies in particular to subquotients of localized extended Weyl algebras $\mathcal{A}_{r,n-r}=k[x_1,\ldots,x_r,x_{r+1}^{\pm1}, \ldots, x_{n}^{\pm1},\partial_1,\ldots,\partial_n],$ where it can be made explicit in terms of convex geometry. We recall these results and then turn to the corresponding primitive quotients and study their Goldie ranks. We prove that the primitive quotients fall into finitely many families whose Goldie ranks are given by a common quasi-polynomial and then realize these quasi-polynomials as Ehrhart quasi-polynomials arising from convex geometry.

We exhibit seven linear codes exceeding the current best known minimum distance $d$ for their dimension $k$ and block length $n$ . Each code is defined over ${ \mathbb{F} }_{8} $ , and their invariants $[n, k, d] $ are given by $[49, 13, 27] $ , $[49, 14, 26] $ , $[49, 16, 24] $ , $[49, 17, 23] $ , $[49, 19, 21] $ , $[49, 25, 16] $ and $[49, 26, 15] $ . Our method includes an exhaustive search of all monomial evaluation codes generated by points in the $[0, 5] \times [0, 5] $ lattice square.

In this note we give a brief review of the construction of a toric variety $\mathcal{V}$ coming from a genus $g\,\ge \,2$ Riemann surface ${{\sum }^{g}}$ equipped with a trinion, or pair of pants, decomposition. This was outlined by J. Hurtubise and L. C. Jeffrey. A. Tyurin used this construction on a certain collection of trinion decomposed surfaces to produce a variety $D{{M}_{g,}}$ the so-called Delzant model of moduli space, for each genus $g$ . We conclude this note with some basic facts about the moment polytopes of the varieties $\mathcal{V}$ . In particular, we show that the varieties $D{{M}_{g}}$ constructed by Tyurin, and claimed to be smooth, are in fact singular for $g\,\ge \,3$ .