The Expected Depth of Random Real Algebraic Plane Curves

Abstract

In this note we study asymptotic isotopy of random real algebraic plane curves. More precisely, we obtain a Kac-Rice type formula that gives the expected number of two-sided components (i.e.\ ovals) of a random real algebraic plane curve winding around a given point. In particular, we show that expected number of such ovals for an even degree Kostlan polynomial is $\frac{\sqrt{d}}{2}$ and independent of the given point.

This paper is withdrawn due to an error in the proof of the lower bound on the expected depth. In the subsequent work [arXiv:2604.18350], we establish by different methods that the expected depth tends to infinity with the degree d, with a lower bound of order O(log(log(d))) and the same upper bound as stated here

References in corpus (4)

• Topological properties of real algebraic varieties: du côté de chez Rokhlin
• Betti numbers of random real hypersurfaces and determinants of random symmetric matrices
• On the number of connected components of random algebraic hypersurfaces
• Expected volume and Euler characteristic of random submanifolds