实代数曲线的双切线：符号计数与构造
Bitangents of real algebraic curves: signed count and constructions
AI总结：
本文证明了实代数曲线双切线的符号计数仅依赖于实拓扑类型，从而得出偶次曲线至少有 d(d-2)/2 条实双切线，并通过构造给出最佳数量。
摘要：
我们从两个角度研究实代数平面曲线的实双切线。首先，我们证明存在一个仅依赖于曲线的实拓扑类型的双切线符号计数。由此可知，偶次度 $d$ 的一般实代数曲线至少有 $\frac{d(d-2)}{2}$ 条实双切线。其次，我们解释如何定位 $\mathbb{C}P^2$ 中多重实圆锥的实扰动（实）双切线。作为主要应用，我们展示了一个具有总计 $318$ 条实双切线和 $6$ 条复双切线的实六次曲线，并进行渐近构造，据我们所知，这给出了给定次数的实代数平面曲线实双切线的最佳数量。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
