On amoebas of random plane curves

27th Gökova Geometry-Topology Conference, Muğla, Turkey, 30 May - 04 June 2022, pp.1

Publication Type: Conference Paper / Summary Text
City: Muğla
Country: Turkey
Page Numbers: pp.1
Middle East Technical University Affiliated: Yes

Abstract

Due to a theorem of Passare and Rullgard, the area of the amoeba of a degree d" role="presentation" >d $d$ algebraic curve in the complex projective plane is bounded above by π2d2/2" role="presentation" >π2d2/2 $π^{2} d^{2} / 2$ and the curves attaining the bound - special Harnack curves - have been characterized by Mikhalkin. In this talk, reporting on joint work with Turgay Bayraktar, I will argue that the expected area of a randomly chosen complex algebraic curve, with respect to the Kostlan distribution, is bounded above by a constant times d" role="presentation" >d $d$ . This result also generalizes in a natural way to half dimensional complete intersections in toric varieties with an arbitrary Newton polytope.