Related papers: The merging operation and $(d-i)$-simplicial $i$-s…

Let $\Phi$ be an irreducible (possibly noncrystallographic) root system of rank $l$ of type $P$. For the corresponding cluster complex $\Delta(P)$, which is known as pure $(l-1)$-dimensional simplicial complex, we define the generating…

Neighborly cubical polytopes exist: for any $n\ge d\ge 2r+2$, there is a cubical convex d-polytope $C^n_d$ whose $r$-skeleton is combinatorially equivalent to that of the $n$-dimensional cube. This solves a problem of Babson, Billera &…

Let $\phi$ be a birational map of the complex projective plane. We know that $\phi$ can be written as a composition of automorphisms of $\mathbb{P}^2_\mathbb{C}$ and the standard quadratic birational map $\sigma$. This writing, that is…

We introduce a new algebraic construction, {\em monop}, that combines monoids (with respect to the product of species), and operads (monoids with respect to the substitution of species) in the same algebraic structure. By the use of…

The combinatorial structure of a d-dimensional simple convex polytope can be reconstructed from its abstract graph [Blind & Mani 1987, Kalai 1988]. However, no polynomial/efficient algorithm is known for this task, although a polynomially…

Let $V$ be a real vector space of dimension $n$ and let $M\subset V$ be a lattice. Let $P\subset V$ be an $n$-dimensional polytope with vertices in $M$, and let $\varphi\colon V\rightarrow \CC $ be a homogeneous polynomial function of…

A $d$-dimensional closed convex set $K$ in $\mathbb{R}^d$ is said to be lattice-free if the interior of $K$ is disjoint with $\mathbb{Z}^d$. We consider the following two families of lattice-free polytopes: the family $\mathcal{L}^d$ of…

This short article serves as the appendix for [Tran and Wang, 2022]. We prove that a complete intersection of $n$ generic polyhedral hypersurfaces in $\mathbb{R}^d$ is $(d-n-1)$-connected for $d\geq 2, d>n$.

We show that by cutting off the vertices and then the edges of neighborly cubical polytopes, one obtains simple 4-dimensional polytopes with n vertices such that all separators of the graph have size at least $\Omega(n/\log^{3/2}n)$. This…

Let $S$ be a $k$-colored (finite) set of $n$ points in $\mathbb{R}^d$, $d\geq 3$, in general position, that is, no {$(d + 1)$} points of $S$ lie in a common $(d - 1)$}-dimensional hyperplane. We count the number of empty monochromatic…

We construct simple geometric operations on faces of the Cayley sum of two polytopes. These operations can be thought of as convex geometric counterparts of divided difference operators in Schubert calculus. We show that these operations…

We construct a 2-parameter family of 4-dimensional polytopes with extreme combinatorial structure: In this family, the ``fatness'' of the f-vector gets arbitrarily close to 9, the ``complexity'' (given by the flag vector) gets arbitrarily…

A quadrangulation is a graph embedded on the sphere such that each face is bounded by a walk of length 4, parallel edges allowed. All quadrangulations can be generated by a sequence of graph operations called vertex splitting, starting from…

We give an explicit construction of vertex-transitive tight triangulations of $d$-manifolds for $d\geq 2$. More explicitly, for each $d\geq 2$, we construct two $(d^2+5d+5)$-vertex neighborly triangulated $d$-manifolds whose vertex-links…

Let $D_\lambda^{d,k}$ denote the family of diagonal hypersurface over a finite field $\mathbb{F}_q$ given by \begin{align*} D_\lambda^{d,k}:X_1^d+X_2^d=\lambda dX_1^kx_2^{d-k}, \end{align*} where $d\geq2$, $1\leq k\leq d-1$, and…

We prove that there exist four operations such that given any two strongly $2$-connected digraphs $H$ and $D$ where $H$ is a butterfly-minor of $D$, there exists a sequence $D_0,\dots, D_n$ where $D_0=H$, $D_n=D$ and for every $0\leq i\leq…

A new generalization of multiquadric functions $\phi(x)=\sqrt{c^{2d}+||x||^{2d}}$, where $x\in\mathbb{R}^n$, $c\in \mathbb{R}$, $d\in \mathbb{N}$, is presented to increase the accuracy of quasi-interpolation further. With the restriction to…

In this paper we present a new technique to construct neighborly polytopes, and use it to prove a lower bound of ((r+d)^((r/2+d/2)^2))/(r^((r/2)^2)d^((d/2)^2)e^(3rd/4)) for the number of combinatorial types of vertex-labeled neighborly…

We give an explicit construction, based on Hadamard matrices, for an infinite series of floor{sqrt{d}/2}-neighborly centrally symmetric d-dimensional polytopes with 4d vertices. This appears to be the best explicit version yet of a recent…

Given two families $X$ and $Y$ of integral polytopes with nice combinatorial and algebraic properties, a natural way to generate new class of polytopes is to take the intersection $\mathcal{P}=\mathcal{P}_1\cap\mathcal{P}_2$, where…