Related papers: Computational Geometry Column 32

In this paper, we prove the first incidence bound for points and conics over prime fields. As applications, we prove new results on expansion of bivariate polynomial images and on certain variations of distinct distances problems. These…

We show that the problem of deciding whether a knot in a fixed closed orientable 3-dimensional manifold bounds a surface of genus at most $g$ is in co-NP. This answers a question of Agol, Hass, and Thurston in 2002. Previously, this was…

We study the local profiles of trees. We show that, in contrast with the situation for general graphs, the limit set of k-profiles of trees is convex. We initiate a study of the defining inequalities of this convex set. Many challenging…

We prove boundedness of rationally-connected threefolds in $\mathbb P^6$ under some extra-assumptions.

We study the family of irreducible curves with $\delta$ nodes belonging to a free linear system $|C|$ with smooth general member on a surface $S$ such that $|K_S|$ is ample. Under the assumption that $C$ is numerically equivalent to $pK_S$,…

Building on work by Dan-Cohen--Wewers, Dan-Cohen [DC], and Brown, we push the computational boundary of our explicit motivic version of Kim's method in the case of the thrice punctured line over an open subscheme of Spec ZZ. To do so, we…

We prove that, for all $k \ge 3,$ and any integers $\Delta, n$ with $n \ge \Delta,$ there exists a $k$-uniform hypergraph on $n$ vertices with maximum degree at most $\Delta$ whose $4$-color Ramsey number is at least $\mathrm{tw}_k(c_k…

A problem from Democritus is used to illustrate the building, and use, of infinitesimal covectors from its regularized, finite, counterpart.

A collection $ \Delta $ of simple closed curves on an orientable surface is an algebraic $ k $-system if the algebraic intersection number $\langle \alpha,\beta \rangle$ is equal to $k $ in absolute value for every $ \alpha , \beta \in…

We prove that the number of directions contained in a set of the form $A \times B \subset AG(2,p)$, where $p$ is prime, is at least $|A||B| - \min\{|A|,|B|\} + 2$. Here $A$ and $B$ are subsets of $GF(p)$ each with at least two elements and…

In this paper I present a new method of studying the densities of the Collatz trajectories generated by a set $S \subset \mathbb{N}$. This method is used to furnish an alternative proof that $d(\{y \in \mathbb{N} : \exists k \text{ where }…

We introduce a new infinite family of regular graphs admitting nested solutions in the edge-isoperimetric problem for all their Cartesian powers. The obtained results include as special cases most of previously known results in this area.

This is an appendix to the paper {\bf Asymptotic K-theory for groups acting on $\tA_2$ buildings}, and contains the results of the computations performed by the authors.

Let M be a hyperbolic 3-manifold with nonempty totally geodesic boundary. We prove that there are upper and lower bounds on the diameter of the skinning map of M that depend only on the volume of the hyperbolic structure with totally…

We extend an earlier result by Dan Abramovich, showing that a conjecture of S. Lang's implies the existence of a uniform bound on the number of $K$-rational points over all smooth curves of genus $g$ defined over $K$, where $K$ is any…

In this paper, we use semidefinite programming and representation theory to compute new lower bounds on the crossing number of the complete bipartite graph $K_{m,n}$, extending a method from de Klerk et al. [SIAM J. Discrete Math. 20…

We prove the lower bound R(M_m) \geq 3/2 m^2 - 2 on the border rank of m x m matrix multiplication by exhibiting explicit representation theoretic (occurence) obstructions in the sense of the geometric complexity theory (GCT) program. While…

In this paper, we prove a version of the Colored Tverberg Theorem with new constraints on the faces, in which we limit the number of faces with each one of the colors.

We prove that a set of finite perimeter is indecomposable if and only if it is, up to a choice of suitable representative, connected in the 1-fine topology. This gives a topological characterization of indecomposability which is new even in…

We prove some new bounds for the size of the maximal dissociated subset of structured (having small sumset, large energy and so on) subsets A of an abelian group.