Related papers: Pyramids and monomial blowing-ups

Let $R$ be a ring and $B = R[X_1, \dots, X_n]$ the polynomial ring in $n$ variables over $R$. In this article, we consider retractions $\varphi : B \longrightarrow B$ such that $\varphi(X_i)$ is either a monic monomial or $0$. We prove that…

An $n\times n$ symmetric matrix $A$ is copositive if the quadratic form $x^TAx$ is nonnegative on the nonnegative orthant. The cone of copositive matrices strictly contains the cone of completely positive matrices, i.e., all matrices of the…

The Topological Tverberg Theorem claims that any continuous map of a (q-1)(d+1)-simplex to \R^d identifies points from q disjoint faces. (This has been proved for affine maps, for d=1, and if q is a prime power, but not yet in general.) The…

To a simplicial complex, we associate a square-free monomial ideal in the polynomial ring generated by its vertex set over a field. We study algebraic properties of this ideal via combinatorial properties of the simplicial complex. By…

The type A_n full root polytope is the convex hull in R^{n+1} of the origin and the points e_i-e_j for 1<= i<j <= n+1. Given a tree T on the vertex set [n+1], the associated root polytope P(T) is the intersection of the full root polytope…

We consider pyramids made of one-dimensional pieces of fixed integer length a and which may have pairwise overlaps of integer length from 1 to a. We prove that the number of pyramids of size m, i.e. consisting of m pieces, equals (am-1,m-1)…

Given a finite collection P of convex n-polytopes in RP^n (n>1), we consider a real projective manifold M which is obtained by gluing together the polytopes in P along their facets in such a way that the union of any two adjacent polytopes…

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…

Every $n th$ order monic polynomial corresponds $n$-dimensional vector. If the given polynomial is stable that is all its roots lie in the open left half plane it is said to be Hurwitz polynomial and the corresponding vector is called…

A planar polygonal billiard $\P$ is said to have the finite blocking property if for every pair $(O,A)$ of points in $\P$ there exists a finite number of ``blocking'' points $B_1, ..., B_n$ such that every billiard trajectory from $O$ to…

We investigate completed interlacing of zeros for pairs of polynomial sequences that fail to interlace by exactly two points. Using a general mixed recurrence relation, we identify a quadratic polynomial whose zeros serve as the two extra…

In the first part of the paper, we give an explicit algorithm to compute the (genus zero) Gromov-Witten invariants of blow-ups of an arbitrary convex projective variety in some points if one knows the Gromov-Witten invariants of the…

We classify when the blowup of a complex Grassmannian $G(k, n)$ along a smooth Schubert subvariety $Z$ is Fano. We compute almost all the two-point, genus zero Gromov-Witten invariants of the blowup when $Z=G(k, n-1)$. We further prove a…

The blow-up of a graph is obtained by replacing every vertex with a finite collection of copies so that the copies of two vertices are adjacent if and only if the originals are. If every vertex is replaced with the same number of copies,…

We prove the theorem mentioned in the title, for ${\mathbb{R}}^n$, where $n \ge 3$. The case of the simplex was known previously. Also, the case $n=2$ was settled, but there the infimum was some well-defined function of the side lengths. We…

Many examples of nonpositively curved closed manifolds arise as blow-ups of projective hyperplane arrangements. If the hyperplane arrangement is associated to a finite reflection group W, and the blow-up locus is W-invariant, then the…

Suppose that $\Phi:X\to Y$ is a morphism from a 3 fold to a surface (over an algebraically closed field of characteristic zero). We prove that there exist sequences of blowups of nonsingular subvarieties $X_1\to X$ and $Y_1\to Y$ such that…

We introduce a Bernstein-Bezier basis for the pyramid, whose restriction to the face reduces to the Bernstein-Bezier basis on the triangle or quadrilateral. The basis satisfies the standard positivity and partition of unity properties…

Given a planar polynomial vector field $X$ with a fixed Newton polytope $\mathcal{P}$, we prove (under some non degeneracy conditions) that the monomials associated to the upper boundary of $\mathcal{P}$ determine (under topological…

We show that a nontrivial graph isomorphism problem of two undirected graphs, and more generally, the permutation similarity of two given $n\times n$ matrices, is equivalent to equalities of volumes of the induced three convex bounded…