Related papers: Every rational polyhedron has finite split rank: n…
In this paper, we provide an equivalent condition for the Chvatal-Gomory (CG) closure of a closed convex set to be finitely-generated. Using this result, we are able to prove that, for any closed convex set that can be written as the…
A rational function of degree at least two with coefficients in an algebraically closed field is post-critically finite (PCF) if all of its critical points have finite forward orbit under iteration. We show that the collection of PCF…
The strength of a multivariate homogeneous polynomial is the minimal number of terms in an expression as a sum of products of lower-degree homogeneous polynomials. Partition rank is the analogue for multilinear forms. Both ranks can drop…
Let $X$ be a cubic fourfold in $P^5_{C}$. We prove that, assuming the Hodge conjecture for the product $S \times S$, where $S$ is a complex surface, and the finite dimensionality of the Chow motive $h(S)$, there are at most a countable…
Let $P$ be a rational polyhedron in $\mathbb{R}^d$ and let $\mathcal{L}$ be a class of $d$-dimensional maximal lattice-free rational polyhedra in $\mathbb{R}^d$. For $L \in \mathcal{L}$ by $R_L(P)$ we denote the convex hull of points…
A split of a polytope $P$ is a (regular) subdivision with exactly two maximal cells. It turns out that each weight function on the vertices of $P$ admits a unique decomposition as a linear combination of weight functions corresponding to…
We present experimental evidence to support the widely held belief that one half of all elliptic curves have infinitely many rational points. The method used to gather this evidence is a refinement of an algorithm due to the author which is…
Brjuno and R\"ussmann proved that every irrationally indifferent fixed point of an analytic function with a Brjuno rotation number is linearizable, and Yoccoz proved that this is sharp for quadratic polynomials. Douady conjectured that this…
Recently it was shown that it is undecidable whether a term rewrite system can be proved terminating by a polynomial interpretation in the natural numbers. In this paper we show that this is also the case when restricting the…
The $\{0,\frac{1}{2}\}$-closure of a rational polyhedron $\{ x \colon Ax \le b \}$ is obtained by adding all Gomory-Chv\'atal cuts that can be derived from the linear system $Ax \le b$ using multipliers in $\{0,\frac{1}{2}\}$. We show that…
Recently, cutting planes derived from maximal lattice-free convex sets have been studied intensively by the integer programming community. An important question in this research area has been to decide whether the closures associated with…
B. Blackadar recently proved that any full corner $pAp$ in a unital C*-algebra $A$ has K-theoretic stable rank greater than or equal to the stable rank of $A$. (Here $p$ is a projection in $A$, and fullness means that $ApA=A$.) This result…
In a finite-dimensional real vector space furnished with a rational structure with respect to a subfield of the field of real numbers, every (simplicial) rational semifan is contained in a complete (simplicial) rational semifan. In this…
The main purpose of this paper is to prove that the positive real numbers can be decomposed into finitely many disjoint pieces which are also closed under addition and multiplication. As a byproduct of the argument we determine all the…
We consider elliptic surfaces whose coefficients are degree $2$ polynomials in a variable $T$. It was recently shown that for infinitely many rational values of $T$ the resulting elliptic curves have rank at least $1$. In this article, we…
A main purpose of this paper is to prove that the class of finite dimensional algebras which verify Han's conjecture is closed under split bounded extensions.
Given an integral polyhedron P and a rational polyhedron Q living in the same n-dimensional space and containing the same integer points as P, we investigate how many iterations of the Chv\'atal-Gomory closure operator have to be performed…
We prove that any finite-degree polynomial functor is topologically Noetherian. This theorem is motivated by the recent resolution of Stillman's conjecture and a recent Noetherianity proof for the space of cubics. Via work by…
The positive semidefinite rank of a convex body $C$ is the size of its smallest positive semidefinite formulation. We show that the positive semidefinite rank of any convex body $C$ is at least $\sqrt{\log d}$ where $d$ is the smallest…
We prove that every three-dimensional polyhedron is uniquely determined by its dihedral angles and edge lengths, even if nonconvex or self-intersecting, under two plausible sufficient conditions: (i) the polyhedron has only convex faces and…