Séminaire Géométries ICJ

Amoebas of random half-dimensional complete intersections

by Özgür Kişisel

112 (ICJ)



1er étage bâtiment Braconnier, Université Claude Bernard Lyon 1 - La Doua
Due to a theorem of Passare and Rullg\aa rd, the area of the amoeba of a degree $d$ algebraic curve in the complex projective plane is bounded above by $\pi^2 d^2/2$. A theorem of Mikhalkin generalizes this result to half-dimensional complete intersections as follows: Let $V$ be a complete intersection of hypersurfaces of degrees $d_{1}, d_{2}, \ldots, d_{n}$ in $\mathbb{CP}^{n}$ and let $\mathcal{A}(V)$ denote its amoeba. Then $Vol(\mathcal{A}(V))\leq \pi^{2n}(d_{1}d_{2}\ldots d_{n})^{2}/2$. My goal in this talk will be to show that if the defining polynomials of the complete intersection are chosen at random with respect to Kostlan distribution, then there exists a constant $c$ independent of the $d_{i}$'s such that the expected volume of the amoeba satisfies the inequality $\mathbb{E}(\mathcal{A}(V))\leq cd_{1}d_{2}\ldots d_{n}$. This result generalizes to other Newton polytopes as well. This is joint work with Turgay Bayraktar.