Related papers: Polytopes of Minimum Positive Semidefinite Rank

We show that a $d$-dimensional polyhedron $S$ in $\real^d$ can be represented by $d$-polynomial inequalities, that is, $S = \{x \in \real^d : p_0(x) \ge 0, >..., p_{d-1}(x) \ge 0 \}$, where $p_0,...,p_{d-1}$ are appropriate polynomials.…

We analyze self-dual polyhedral cones and prove several properties about their slack matrices. In particular, we show that self-duality is equivalent to the existence of a positive semidefinite (PSD) slack. Beyond that, we show that if the…

The minimum rank of a graph is the minimum possible rank of a real matrix whose zero-nonzero pattern is described by the graph. The current algorithms can compute efficiently the minimum rank of undirected trees. This paper provides an…

The rank of a finite semigroup is the smallest number of elements required to generate the semigroup. A formula is given for the rank of an arbitrary (non necessarily regular) Rees matrix semigroup over a group. The formula is expressed in…

For a $d$-dimensional polytope with $v$ vertices, $d+1\le v\le2d$, we calculate precisely the minimum possible number of $m$-dimensional faces, when $m=1$ or $m\ge0.62d$. This confirms a conjecture of Gr\"unbaum, for these values of $m$.…

In this article, we investigate the rank index of projective curves $\mathscr{C} \subset \mathbb{P}^r$ of degree $r+1$ when $\mathscr{C} = \pi_p (\tilde{\mathscr{C}})$ for the standard rational normal curve $\tilde{\mathscr{C}} \subset…

The structure of all graphs having minimum rank at most k over a finite field with q elements is characterized for any possible k and q. A strong connection between this characterization and polarities of projective geometries is explained.…

The minimum rank of a graph G is the minimum rank over all real symmetric matrices whose off-diagonal sparsity pattern is the same as that of the adjacency matrix of G. In this note we present the first exact algorithm for the minimum rank…

We prove a lower bound theorem for the number of $k$-faces ($1\le k\le d-2$) in a $d$-dimensional polytope $P$ (or $d$-polytope) with up to $3d-1$ vertices. Previous lower bound theorems for $d$-polytopes with few vertices concern those…

A polynomial $p\in\mathbb{R}[x]$ is a divisor of some polynomial $0\neq f\in\mathbb{R}[x]$ with non-negative coefficients if and only if $p$ does not have a positive real root. The lowest possible degree of such $f$ for a given $p$ is known…

We study the closure of the projection of the (nonconvex) cone of rank restricted positive semidefinite matrices onto subsets of the matrix entries. This defines the feasible sets for semidefinite completion problems with restrictions on…

The permanent vs. determinant problem is one of the most important problems in theoretical computer science, and is the main target of geometric complexity theory proposed by Mulmuley and Sohoni. The current best lower bound for the…

A cubical polytope is a polytope with all its facets being combinatorially equivalent to cubes. We deal with the connectivity of the graphs of cubical polytopes. We first establish that, for any $d\ge 3$, the graph of a cubical $d$-polytope…

Let $\sigma_b(X_{m,d}(\mathbb {C}))(\mathbb {R})$, $b(m+1) < \binom{m+d}{m}$, denote the set of all degree $d$ real homogeneous polynomials in $m+1$ variables (i.e. real symmetric tensors of format $(m+1)\times ... \times (m+1)$, $d$ times)…

We define the \emph{curvilinear rank} of a degree $d$ form $P$ in $n+1$ variables as the minimum length of a curvilinear scheme, contained in the $d$-th Veronese embedding of $\mathbb{P}^n$, whose span contains the projective class of $P$.…

The rank of a semigroup is the cardinality of a smallest generating set. In this paper we compute the rank of the endomorphism monoid of a non-trivial uniform partition of a finite set, that is, the semigroup of those transformations of a…

An extended formulation of a polytope P is a polytope Q which can be projected onto P. Extended formulations of small size (i.e., number of facets) are of interest, as they allow to model corresponding optimization problems as linear…

Fractional minimum positive semidefinite rank is defined from $r$-fold faithful orthogonal representations and it is shown that the projective rank of any graph equals the fractional minimum positive semidefinite rank of its complement. An…

A weakly infeasible semidefinite program (SDP) has no feasible solution, but it has approximate solutions whose constraint violation is arbitrarily small. These SDPs are ill-posed and numerically often unsolvable. They are also closely…

This article announces the completion of the classification of rank 4 locally projective polytopes and their quotients. There are seventeen universal locally projective polytopes (nine nondegenerate). Amongst their 441 quotients are a…