Real Zeuthen numbers for two lines

2007/10/04 by Benoit Bertrand, Benoît Bertrand, Bertrand, Benoit
Computer Science · Engineering · Mathematics · #14N10 #14P99 #Algebraic Geometry (math.AG) #FOS: Mathematics #Mathematics and Applications #Polynomial and algebraic computation #graph theory and CDMA systems #math.AG #msc:14N10 #msc:14P99

paper · pdf · doi:10.48550/arxiv.0710.1095

6 pages, 3 figures

arxiv created 2007/10/04 · openalex publication_date 2007/10/04 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

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 \PP². 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.

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