摘要翻译：
给定三个自然数$k，l，d$,使得$k+l=d(d+3)/2$,Zeuthen数$n_{d}(l)$是$\pp^2$中经过$k$点并与$l$线相切的$d$次的非奇异复代数曲线的个数。它不依赖于所选点和线的一般配置$C$。如果点和线是实的，那么实曲线的相应数目$n_{d}^\rr(l,C)$通常取决于所选择的配置。利用Mikhalkin的热带对应定理证明了对于两条直线，实Zeuthen问题是极大的：存在一个构型$C$，使得$N_{d}^\rr(2，C)=N_{d}(2)$。对应定理将计算简化为用多重数计算某些格路径。
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英文标题：
《Real Zeuthen numbers for two lines》
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作者：
Benoit Bertrand
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最新提交年份：
2007
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分类信息：

一级分类：Mathematics        数学
二级分类：Algebraic Geometry        代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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英文摘要：
  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.
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PDF链接：
https://arxiv.org/pdf/0710.1095