Bounding the signed count of real bitangents to plane quartics (Q6120307)

In this article, the authors prove a conjecture by Larson and Vogt regarding the signed count of the number of real bitangents to real smooth plane quartics. Definition. Let \(Q\) be a quartic curve in \(\mathbb{P}^2\), \(B\) be a bitangent line to \(Q\), and \(Q\cap B = 2Z\), where \(Z = z_1+z_2\) is a degree two divisor. Suppose \(L\) is a real line such that \(L\) is disjoint from the points \(z_1\) and \(z_2\). (Such a line \(L\) is called \textit{admissible}). Denote by \(\partial_L\) a derivation with respect to a linear form vanishing along \(L\). The \textit{QType of \(B\) with respect to \(L\)} is defined as: \[\operatorname{QType}_L(B) := \operatorname{sign}(\partial_L f(z_1) \cdot \partial_L f(z_2)))\in \{+1,-1\}.\] For an admissible line \(L\subset \mathbb{P}^2\), \[ s_L(Q):= \sum_{B \text{ real bitangent}} \operatorname{QType}_L(B)\] \textit{H. Larson} and \textit{I. Vogt} [Res. Math. Sci. 8, No. 2, Paper No. 26, 21 p. (2021; Zbl 1471.14070)] proved that \(s_L(Q)\) is nonnegative even integers and conjectured that \(s_L(Q)\) is at most \(8\). The authors prove that \(s_L(Q)\in\{0,2,4,6,8\}\) by gathering ideas from real algebraic geometry and plane geometry.