Amoebas of random half-dimensional complete intersections

par Özgür Kişisel

vendredi 23 sept. 2022, 10:30 → 11:30 Europe/Paris

112 (ICJ)

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 $d_1,d_2,\dots ,d_n$ in ${\mathbb{CP}}^n$ and let $\mathcal{A}(V)$ denote its amoeba. Then $Vol(\mathcal{A}(V))\le {\pi }^{2n}(d_1{d}_2\dots d_n{)}^2/2$. 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 $\mathbb{E}(\mathcal{A}(V))\le c{d}_1{d}_2\dots d_n$. This result generalizes to other Newton polytopes as well. This is joint work with Turgay Bayraktar.