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Real Zeuthen numbers for two lines

Published in International Mathematics Research Notices. 2008, vol. 8, no. ID rnn014, p. 8 p.
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.
arXiv: 0710.1095
Note 6 pages, 3 figures
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BERTRAND, Benoît. Real Zeuthen numbers for two lines. In: International Mathematics Research Notices, 2008, vol. 8, n° ID rnn014, p. 8 p. doi: 10.1093/imrn/rnn014 https://archive-ouverte.unige.ch/unige:8541

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Deposited on : 2010-07-02

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