The Expected Depth of Random Real Algebraic Plane Curves
Algebraic Geometry
2026-04-22 v2 Complex Variables
Probability
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 and independent of the given point.
Keywords
Cite
@article{arxiv.2110.03198,
title = {The Expected Depth of Random Real Algebraic Plane Curves},
author = {Turgay Bayraktar and Ali Ulaş Özgür Kişisel},
journal= {arXiv preprint arXiv:2110.03198},
year = {2026}
}
Comments
This paper is withdrawn due to an error in the proof of the lower bound on the expected depth.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