Amoeba Measures of Random Plane Curves

Abstract

We prove that the expected area of the amoeba of a complex plane curve of degree $d$ is less than $3\ln (d{)}^2/2+9\ln (d)+9$ and once rescaled by $\ln (d{)}^2$, is asymptotically bounded from below by $3/4$. In order to get this lower bound, given disjoint isometric embeddings of a bidisc of size $1/\sqrt{d}$ in the complex projective plane, we lower estimate the probability that one of them is a submanifold chart of a complex plane curve. It exponentially converges to one as the number of bidiscs grow to $+\infty$.

41 pages, 1 figure