Related papers: Quadratic Gr\"obner bases arising from partially o…

The first part of this paper concerns type C. We present new explicitly defined families of algebro-combinatorial structures of three kinds: combinatorial bases in representations, Newton--Okounkov bodies of flag varieties and toric…

Let $P$ be a finite poset, $K$ a field, and $O(P)$ (resp. $C(P)$) the order (resp. chain) polytope of $P$. We study the non-Gorenstein locus of $E_K[O(P)]$ (resp. $E_K[C(P)]$), the Ehrhart ring of $O(P)$ (resp. $C(P)$) over $K$, which are…

Let $\Gamma=(X,\mathcal{R})$ denote a finite, simple, connected, and undirected non-bipartite graph with vertex set $X$ and edge set $\mathcal{R}$. Fix a vertex $x \in X$, and define $\mathcal{R}_f = \mathcal{R} \setminus \{yz \mid…

We investigate arithmetic, geometric and combinatorial properties of symmetric edge polytopes. We give a complete combinatorial description of their facets. By combining Gr\"obner basis techniques, half-open decompositions and methods for…

Let $G$ denote a $Q$-polynomial distance-regular graph with diameter $D$ at least 4. Assume that the intersection numbers of $G$ satisfy $a_i=0$ for $0 \leq i \leq D-1$ and $a_D\neq 0$. We show that $G$ is a polygon, a folded cube, or an…

We study C*-algebras associated to right LCM semigroups, that is, semigroups which are left cancellative and for which any two principal right ideals are either disjoint or intersect in another principal right ideal. If $P$ is such a…

Quadratically constrained quadratic programs (QCQPs) are a highly expressive class of nonconvex optimization problems. While QCQPs are NP-hard in general, they admit a natural convex relaxation via the standard (Shor) semidefinite program…

The true prosoluble completion $P\Cal S (\Gamma)$ of a group $\Gamma$ is the inverse limit of the projective system of soluble quotients of $\Gamma$. Our purpose is to describe examples and to point out some natural open problems. We…

Preorder polytopes, defined from preorders on finite sets, are introduced and studied from a lattice point enumeration point of view. They naturally generalize arbor polytopes, recently introduced and studied by the second named author.…

For a graph $G$ with $p$ vertices the closed convex cone $\mathbb{S}^p_{\succeq0}(G)$ consists of all real positive semidefinite $p\times p$ matrices with zeros in the off-diagonal entries corresponding to nonedges of $G$. The extremal rays…

As an instance of the B-polynomial, the circuit, or cycle, polynomial P(G(Gamma); w) of the generalized rooted product G(Gamma) of graphs was studied by Farrell and Rosenfeld ({\em Jour. Math. Sci. (India)}, 2000, \textbf{11}(1), 35--47)…

We determine the squarefree part of the scalar factor that arises when the quartic invariant of the generic binary form $F$ of odd degree $2n+1$ is expressed as the discriminant of the unique quadratic covariant $(F,F)_{2n}$. This…

The extension complexity $\mathsf{xc}(P)$ of a polytope $P$ is the minimum number of facets of a polytope that affinely projects to $P$. Let $G$ be a bipartite graph with $n$ vertices, $m$ edges, and no isolated vertices. Let…

We consider random polynomials of the form $G_n(z):= \sum_{|\alpha|\leq n} \xi^{(n)}_{\alpha}p_{n,\alpha}(z)$ where $\{\xi^{(n)}_{\alpha}\}_{|\alpha|\leq n}$ are i.i.d. (complex) random variables and $\{p_{n,\alpha}\}_{|\alpha|\leq n}$ form…

For right-angled Coxeter groups $W_{\Gamma}$, we obtain a condition on $\Gamma$ that is necessary and sufficient to ensure that $W_{\Gamma}$ is thick and thus not relatively hyperbolic. We show that Coxeter groups which are not thick all…

For $q = p^n$ with $p$ an odd prime, the projective linear group $PGL(2,q)$ can be seen as the stabilizer of a conic $O$ in a projective plane $\pi = PG(2,q)$. In that setting, involutions of $PGL(2,q)$ correspond bijectively to points of…

In this research paper, the problem of optimization of a quadratic form over the convex hull generated by the corners of hypercube is attempted and solved. Some results related to stable states/vectors, anti-stable states/vectors (over the…

Facets of the convex hull of $n$ independent random vectors chosen uniformly at random from the unit sphere in $\mathbb{R}^d$ are studied. A particular focus is given on the height of the facets as well as the expected number of facets as…

We define certain natural finite sums of $n$'th roots of unity, called $G_P(n)$, that are associated to each convex integer polytope $P$, and which generalize the classical $1$-dimensional Gauss sum $G(n)$ defined over $\mathbb Z/ {n…

The triangle of sorted binomial coefficients $\left\langle {n \atop k} \right\rangle = \binom{n}{\lfloor \frac{n - k}{2} \rfloor}$ for $0 \leq k \leq n$ has appeared several times in recent combinatorial works but has evaded dedicated…