AMOEBA MEASURES OF RANDOM PLANE CURVES

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KİŞİSEL A. U. Ö., Welschinger J.

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, vol.379, no.4, pp.2605-2651, 2026 (SCI-Expanded, Scopus)

  • Publication Type: Article / Article
  • Volume: 379 Issue: 4
  • Publication Date: 2026
  • Doi Number: 10.1090/tran/9538
  • Journal Name: TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, MathSciNet, zbMATH, DIALNET
  • Page Numbers: pp.2605-2651
  • Middle East Technical University Affiliated: Yes

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/ root 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 grows to +infinity.