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Amoeba Measures of Random Plane Curves

Ali Ulaş Özgür Kişisel, Jean-Yves Welschinger

Abstract

We prove that the expected area of the amoeba of a complex plane curve of degree $d$ is less than $\displaystyle{3\ln(d)^2/2+9\ln(d)+9}$ 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 $+\infty$.

Amoeba Measures of Random Plane Curves

Abstract

We prove that the expected area of the amoeba of a complex plane curve of degree is less than and once rescaled by , is asymptotically bounded from below by . In order to get this lower bound, given disjoint isometric embeddings of a bidisc of size 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 .
Paper Structure (21 sections, 51 theorems, 207 equations, 1 figure)

This paper contains 21 sections, 51 theorems, 207 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hermitian matrices
  4. Conditional expectation
  5. Gaussian complex vectors
  6. Fubini-Study metric
  7. Bergman kernel
  8. Amoeba measures and upper bounds
  9. Amoeba measures
  10. Level set of the area functional and asymptotic upper estimates
  11. Fubini-Study Charts of Plane Curves
  12. Fubini-Study Charts
  13. Linear Polynomials
  14. An Effective Criterion
  15. The Ball-Average Norm
  16. ...and 6 more sections

Key Result

Theorem 1.1

For all $d>0$, ${\mathbb{E}_{d}(Vol(A))\leq \frac{3}{2}\ln(d)^2+9\ln(d)+9}.$ Furthermore, as $d$ tends to infinity,

Figures (1)

  • Figure 1: The domain $\mathcal{H}_{d}$ in $\mathbb{R}^2$.

Theorems & Definitions (68)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Remark 2.4
  • Lemma 2.5
  • Definition 2.6
  • Lemma 2.7
  • ...and 58 more