1,489 research outputs found
Point-curve incidences in the complex plane
We prove an incidence theorem for points and curves in the complex plane.
Given a set of points in and a set of curves with
degrees of freedom, Pach and Sharir proved that the number of point-curve
incidences is . We
establish the slightly weaker bound
on the number of incidences between points and (complex) algebraic
curves in with degrees of freedom. We combine tools from
algebraic geometry and differential geometry to prove a key technical lemma
that controls the number of complex curves that can be contained inside a real
hypersurface. This lemma may be of independent interest to other researchers
proving incidence theorems over .Comment: The proof was significantly simplified, and now relies on the
Picard-Lindelof theorem, rather than on foliation
Arrangements Of Minors In The Positive Grassmannian And a Triangulation of The Hypersimplex
The structure of zero and nonzero minors in the Grassmannian leads to rich
combinatorics of matroids. In this paper, we investigate an even richer
structure of possible equalities and inequalities between the minors in the
positive Grassmannian. It was previously shown that arrangements of equal
minors of largest value are in bijection with the simplices in a certain
triangulation of the hypersimplex that was studied by Stanley, Sturmfels, Lam
and Postnikov. Here we investigate the entire set of arrangements and its
relations with this triangulation. First, we show that second largest minors
correspond to the facets of the simplices. We then introduce the notion of
cubical distance on the dual graph of the triangulation, and study its
relations with the arrangement of t-th largest minors. Finally, we show that
arrangements of largest minors induce a structure of partially ordered sets on
the entire collection of minors. We use the Lam and Postnikov circuit
triangulation of the hypersimplex to describe a 2-dimensional grid structure of
this poset
Arrangements of equal minors in the positive
Abstract. We discuss arrangements of equal minors in totally positive matrices. More precisely, we would like to investigate the structure of possible equalities and inequalities between the minors. We show that arrangements of equals minors of largest value are in bijection with sorted sets, which earlier appeared in the context of alcoved polytopes and Gröbner bases. Maximal arrangements of this form correspond to simplices of the alcoved triangulation of the hypersimplex; and the number of such arrangements equals the Eulerian number. On the other hand, we conjecture and prove in many cases that arrangements of equal minors of smallest value are exactly the weakly separated sets. Weakly separated sets, originally introduced by Leclerc and Zelevinsky, are closely related to the positive Grassmannian and the associated cluster algebra. Résumé. Il s’agit des arrangements des mineurs égaux dans les matrices totalement positives. Plus précisément, nous aimerions étudier la structure des égalités et inégalités possibles entre les mineurs. Nous montrons que les arrangements des mineurs égaux de plus grande valeur sont en bijection avec les ensembles triés, qui auparavant apparaissaient dans le cadre de polytopes alcôve et bases de Gröbner. Arrangements maximales de ce format correspondent aux simplexes de la triangulation alcôve de la hypersimplex, et le nombre de ces arrangements est égal au nombre eulérien. D’autre part, nous conjecturons et prouvons dans des cas nombreux que les arrangements des mineurs égaux de plus petite valeur sont notamment les ensembles faiblement séparés. Ces ensembles faiblement séparés, initialement introduites par Leclerc et Zelevinsky, sont liés à la Grassmannienne positive et l’algèbre cluster
Elation generalised quadrangles of order (s,p), where p is prime
We show that an elation generalised quadrangle which has p+1 lines on each
point, for some prime p, is classical or arises from a flock of a quadratic
cone (i.e., is a flock quadrangle).Comment: 14 page
Sorting orders, subword complexes, Bruhat order and total positivity
In this note we construct a poset map from a Boolean algebra to the Bruhat
order which unveils an interesting connection between subword complexes,
sorting orders, and certain totally nonnegative spaces. This relationship gives
a new proof of Bj\"orner and Wachs' result \cite{BW} that the proper part of
Bruhat order is homotopy equivalent to the proper part of a Boolean algebra ---
that is, to a sphere. We also obtain a geometric interpretation for sorting
orders. We conclude with two new results: that the intersection of all sorting
orders is the weak order, and the union of sorting orders is the Bruhat order.Comment: 10 pages, 1 figure. This is the official version. It is more official
than the version that appears in Advances in Applied Mathematic
N=4 Multi-Particle Mechanics, WDVV Equation and Roots
We review the relation of N=4 superconformal multi-particle models on the
real line to the WDVV equation and an associated linear equation for two
prepotentials, F and U. The superspace treatment gives another variant of the
integrability problem, which we also reformulate as a search for closed flat
Yang-Mills connections. Three- and four-particle solutions are presented. The
covector ansatz turns the WDVV equation into an algebraic condition, for which
we give a formulation in terms of partial isometries. Three ideas for
classifying WDVV solutions are developed: ortho-polytopes, hypergraphs, and
matroids. Various examples and counterexamples are displayed
An Incidence Approach to the Distinct Distances Problem
In 1946, Erdös posed the distinct distances problem, which asks for the minimum number of distinct distances that any set of n points in the real plane must realize. Erdös showed that any point set must realize at least &Omega(n1/2) distances, but could only provide a construction which offered &Omega(n/&radic(log(n)))$ distances. He conjectured that the actual minimum number of distances was &Omega(n1-&epsilon) for any &epsilon \u3e 0, but that sublinear constructions were possible. This lower bound has been improved over the years, but Erdös\u27 conjecture seemed to hold until in 2010 Larry Guth and Nets Hawk Katz used an incidence theory approach to show any point set must realize at least &Omega(n/log(n)) distances. In this thesis we will explore how incidence theory played a roll in this process and expand upon recent work by Adam Sheffer and Cosmin Pohoata, using geometric incidences to achieve bounds on the bipartite variant of this problem. A consequence of our extensions on their work is that the theoretical upper bound on the original distinct distances problem of &Omega(n/&radic(log(n))) holds for any point set which is structured such that half of the n points lies on an algebraic curve of arbitrary degree
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