Real Lines on Random Cubic Surfaces

Abstract

We give an explicit formula for the expectation of the number of real lines on a random invariant cubic surface, i.e. a surface $Z\subset \mathbb{R}{P}^3$ defined by a random gaussian polynomial whose probability distribution is invariant under the action of the orthogonal group $O(4)$ by change of variables. Such invariant distributions are completely described by one parameter $\lambda \in [0,1]$ and as a function of this parameter the expected number of real lines equals: \begin{equation} E_λ=\frac{9(8λ^2+(1-λ)^2)}{2λ^2+(1-λ)^2}\left(\frac{2λ^2}{8λ^2+(1-λ)^2}-\frac{1}{3}+\frac{2}{3}\sqrt{\frac{8λ^2+(1-λ)^2}{20λ^2+(1-λ)^2}}\right). \end{equation} This result generalizes previous results by Basu, Lerario, Lundberg and Peterson for the case of a Kostlan polynomial, which corresponds to $\lambda =\frac{1}{3}$ and for which $E_{\frac{1}{3}}=6\sqrt{2}-3.$ Moreover, we show that the expectation of the number of real lines is maximized by random purely harmonic cubic polynomials, which corresponds to the case $\lambda =1$ and for which $E_1=24\sqrt{\frac{2}{5}}-3$.

References in corpus (6)

• Betti numbers of random real hypersurfaces and determinants of random symmetric matrices
• On the number of connected components of random algebraic hypersurfaces
• Lower estimates for the expected Betti numbers of random real hypersurfaces
• Abundance of real lines on real projective hypersurfaces
• Expected topology of random real algebraic submanifolds
• On the geometry of random lemniscates

Cited by in corpus (2)

• Lines on $p$-adic and real cubic surfaces
• Probabilistic bounds on best rank-one approximation ratio