Real Zeuthen numbers for two lines

Given three natural numbers $k,l,d$ such that $k+l=d(d+3)/2$, the Zeuthen number $N_{d}(l)$ is the number of nonsingular complex algebraic curves of degree $d$ passing through $k$ points and tangent to $l$ lines in \PP^2. It does not depend on the generic configuration $C$ of points and lines chosen. If the points and lines are real, the corresponding number N_{d}^\RR(l,C) of real curves usually depends on the configuration chosen. We use Mikhalkin's tropical correspondence theorem to prove that for two lines the real Zeuthen problem is maximal: there exists a configuration $C$ such that N_{d}^\RR(2,C)=N_{d}(2). The correspondence theorem reduces the computation to counting certain lattice paths with multiplicities.Comment: 6 pages, 3 figure

Euler Characteristic of real nondegenerate tropical complete intersections

We define nondegenerate tropical complete intersections imitating the corresponding definition in complex algebraic geometry. As in the complex situation, all nonzero intersection multiplicity numbers between tropical hypersurfaces defining a nondegenerate tropical complete intersection are equal to 1. The intersection multiplicity numbers we use are sums of mixed volumes of polytopes which are dual to cells of the tropical hypersurfaces. We show that the Euler characteristic of a real nondegenerate tropical complete intersection depends only on the Newton polytopes of the tropical polynomials which define the intersection. Basically, it is equal to the usual signature of a complex complete intersection with same Newton polytopes, when this signature is defined. The proof reduces to the toric hypersurface case, and uses the notion of $E$-polynomials of complex varieties.Comment: Version 1: slight revision of a preprint which appeared on our webpages on April 2007, version 2: abstract expande

A Viro Theorem without convexity hypothesis for trigonal curves

A cumbersome hypothesis for Viro patchworking of real algebraic curves is the convexity of the given subdivision. It is an open question in general to know whether the convexity is necessary. In the case of trigonal curves we interpret Viro method in terms of dessins d'enfants. Gluing the dessins d'enfants in a coherent way we prove that no convexity hypothesis is required to patchwork such curves.Comment: 26 pages, 18 figure

Genus 0 characteristic numbers of the tropical projective plane

Finding the so-called characteristic numbers of the complex projective plane ${\mathbb C}P^2$ is a classical problem of enumerative geometry posed by Zeuthen more than a century ago. For a given $d$ and $g$ one has to find the number of degree $d$ genus $g$ curves that pass through a certain generic configuration of points and at the same time are tangent to a certain generic configuration of lines. The total number of points and lines in these two configurations is $3d-1+g$ so that the answer is a finite integer number. In this paper we translate this classical problem to the corresponding enumerative problem of tropical geometry in the case when $g=0$. Namely, we show that the tropical problem is well-posed and establish a special case of the correspondence theorem that ensures that the corresponding tropical and classical numbers coincide. Then we use the floor diagram calculus to reduce the problem to pure combinatorics. As a consequence, we express genus 0 characteristic numbers of \CC P^2 in terms of open Hurwitz numbers.Comment: 55 pages, 23 figure

Tropical Open Hurwitz numbers

We give a tropical interpretation of Hurwitz numbers extending the one discovered in \cite{CJM}. In addition we treat a generalization of Hurwitz numbers for surfaces with boundary which we call open Hurwitz numbers.Comment: 10 pages, 6 figure

Polynomial systems with few real zeroes

We study some systems of polynomials whose support lies in the convex hull of a circuit, giving a sharp upper bound for their numbers of real solutions. This upper bound is non-trivial in that it is smaller than either the Kouchnirenko or the Khovanskii bounds for these systems. When the support is exactly a circuit whose affine span is ${\Z}^n$, this bound is $2n+1$, while the Khovanskii bound is exponential in $n^2$. The bound $2n+1$ can be attained only for non-degenerate circuits. Our methods involve a mixture of combinatorics, geometry, and arithmetic.Comment: 23 pages, 1 .eps figure. Revised Introductio

Real Zeuthen Numbers for Two Lines

Given three natural numbers k, l, d such that k + l = d(d + 3)/2, the Zeuthen number Nd(l) is the number of nonsingular complex algebraic curves of degree d passing through k points and tangent to l lines in . It does not depend on the generic configuration C of points and lines chosen. If the points and lines are real, the corresponding number of real curves usually depends on the configuration chosen. We use Mikhalkin's tropical correspondence theorem to prove that for two lines, the real Zeuthen problem is maximal: there exists a configuration C such that . The correspondence theorem reduces the computation to counting certain lattice paths with multiplicitie

Asymptotically Maximal Families of Hypersurfaces in Toric Varieties

A real algebraic variety is maximal (with respect to the Smith-Thom inequality) if the sum of the Betti numbers (with $$\mathbb{Z}_2$$ coefficients) of the real part of the variety is equal to the sum of Betti numbers of its complex part. We prove that there exist polytopes that are not Newton polytopes of any maximal hypersurface in the corresponding toric variety. On the other hand we show that for any polytope Δ there are families of hypersurfaces with the Newton polytopes $$(\lambda \Delta )_{\lambda \in \mathbb{N}}$$ that are asymptotically maximal when λ tends to infinity. We also show that these results generalize to complete intersection