52,559 research outputs found
On generic and maximal k-ranks of binary forms
In what follows, we pose two general conjectures about decompositions of
homogeneous polynomials as sums of powers. The first one (suggested by G.
Ottaviani) deals with the generic k-rank of complex-valued forms of any degree
divisible by k in any number of variables. The second one (by the fourth
author) deals with the maximal k-rank of binary forms. We settle the first
conjecture in the cases of two variables and the second in the first
non-trivial case of the 3-rd powers of quadratic binary forms.Comment: 17 pages, 1 figur
On the centralizer of generic braids
We study the centralizer of a braid from the point of view of Garside theory,
showing that generically a minimal set of generators can be computed very efficiently, as the ultra summit set of a generic braid has a very particular structure. We present an algorithm to compute the centralizer of a braid whose generic-case complexity is quadratic on the length of the input, and which outputs a minimal set of generators in the generic case.Ministerio de EconomĂa y CompetitividadFondo Europeo de Desarrollo Regiona
Enhanced negative type for finite metric trees
Finite metric trees are known to have strict 1-negative type. In this paper
we introduce a new family of inequalities that quantify the extent of the
"strictness" of the 1-negative type inequalities for finite metric trees. These
inequalities of "enhanced 1-negative type" are sufficiently strong to imply
that any given finite metric tree must have strict p-negative type for all
values of p in an open interval that contains the number 1. Moreover, these
open intervals can be characterized purely in terms of the unordered
distribution of edge weights that determine the path metric on the particular
tree, and are therefore largely independent of the tree's internal geometry.
From these calculations we are able to extract a new non linear technique for
improving lower bounds on the maximal p-negative type of certain finite metric
spaces. Some pathological examples are also considered in order to stress
certain technical points.Comment: 35 pages, no figures. This is the final version of this paper sans
diagrams. Please note the corrected statement of Theorem 4.16 (and hence
inequality (1)). A scaling factor was omitted in Version #
Implicitization of surfaces via geometric tropicalization
In this paper we further develop the theory of geometric tropicalization due
to Hacking, Keel and Tevelev and we describe tropical methods for
implicitization of surfaces. More precisely, we enrich this theory with a
combinatorial formula for tropical multiplicities of regular points in
arbitrary dimension and we prove a conjecture of Sturmfels and Tevelev
regarding sufficient combinatorial conditions to compute tropical varieties via
geometric tropicalization. Using these two results, we extend previous work of
Sturmfels, Tevelev and Yu for tropical implicitization of generic surfaces, and
we provide methods for approaching the non-generic cases.Comment: 20 pages, 6 figures. Mayor reorganization and exposition improved.
The results on geometric tropicalization have been extended to any dimension.
In particular, Conjecture 2.8 is now Theorem 2.
- …