problems

\(\def\amoeba{\mathcal{A}}\def\Real{\mathbb{R}}\def\Comp{\mathbb{C}}\)

Analytic Combinatorics
Homological independence of natural cycles.

Consider a Laurent polynomial \(P\) in \(d\) variables, and the corresponding amoeba \(\amoeba_P\subset\Real^d\). To each component of the complement to the amoeba (an open convex subset of \(\Real^d\) one can associate a \(d\)-homology class (a natural class) in \(H_d(\Comp_*^d-\{P=0\})\): the class of the tori-preimages of the points in that component.

The problem is to prove that these classes are independent in \(H_d(\Comp_*^d-\{P=0\})\).

It was posed, apparently by Forsberg, Passare, Tsikh in 2000 (and answered affirmatively for the products of linear factors, with the maximal number of components of the complement to the amoeba). It was also proved for Harnack polynomials in \(d=2\) by Lushin and Pochekutov in 2019.


Vertices of Stokes polytopes

For X-shaped Stokes polytopes, the generating function for the number of the vertices is
\[
\sum_{n_1,n_2,n_3,n_4} c_{n_1,n_2,n_3,n_4}z_1^{n_1}z_2^{n_2}z_3^{n_3}z_4^{n_4}=F(z_1,z_2)F(z_2,z_3)F(z_3,z_4)F(z_4,z_1),
\]
where \(c_{n_1,n_2,n_3,n_4}\) is the number of vertices in the Stokes polytope corresponding to the polyomino consisting of the central square with snakes of lengths \(n_1,n_2,n_3,n_4\) attached to its sides, and \[F(z,w)=\frac{\sqrt{1-4z}-\sqrt{1-4w}}{z-w}.\]

What is the asymptotic behavior of \(c_\bf{n}\) if \({\mathbf{n}}\sim n(r_1,r_2,r_3,r_4), n\to\infty\)?

Leave a Reply

Your email address will not be published.