Real Zeuthen numbers for two lines

TL;DR

This paper proves that for two lines, the maximum number of real algebraic curves passing through points and tangent to lines matches the complex count, using tropical geometry techniques.

Contribution

It establishes that the real Zeuthen problem for two lines is maximal, showing the real count can reach the complex number using tropical correspondence.

Findings

Real Zeuthen number for two lines is maximal.

Tropical correspondence theorem reduces the problem to lattice path counting.

Existence of a configuration achieving the maximum real count.

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 $\PP^2$. 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.