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arXiv · 2402.03993

Bitangents of real algebraic curves: signed count and constructions

Thomas Blomme·Erwan Brugallé·Cristhian Garay
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Abstract

We study real bitangents of real algebraic plane curves from two perspectives. We first show that there exists a signed count of such bitangents that only depends on the real topological type of the curve. From this follows that a generic real algebraic curve of even degree $d$ has at least $\frac{d(d-2)}{2}$ real bitangents. Next we explain how to locate (real) bitangents of a (real) perturbation of a multiple (real) conic in $\mathbb{C}P^2$. As main applications, we exhibit a real sextic with a total of $318$ real bitangents and 6 complex ones, and perform asymptotical constructions that give the best, to our knowledge, number of real bitangents of real algebraic plane curves of a given degree.

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Thomas Blomme, Erwan Brugallé, Cristhian Garay. 2024-02-06. Bitangents of real algebraic curves: signed count and constructions. https://arxiv.org/abs/2402.03993

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