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2. Topological problems

    1. Homological independence

          Suppose $f$ is a Laurent polynomial and $V_f \subseteq (\mathbb C^\times)^d$ is the subvariety of the torus. Let $M_f = (\mathbb C^\times)^d \setminus V_f$ and let $A_f$ denote the amoeba of $f$. Let $(C_\omega)_\omega$ denote the components of $\mathbb R^d \setminus A_f$. For each point $\overline \rho \in C_\omega$, let $\mathbb T_{\overline \rho}$ be the torus in $M_f$ that maps to $\overline \rho$ under $\text{Log}$, and let $[\mathbb T_{\omega}]$ denote its homology class in $H_d(M_f)$.

      Problem 2.1.

      [Yuliy Baryshnikov] Are these homology classes $([\mathbb T_\omega])_\omega$ linearly independent?
          The answer is known to be yes in certain special cases – for example if $f$ is a collection of hyperplanes in generic position, or $V_f$ is a Harnack variety.
        • Dual homology basis

              Suppose we have a $d$-dimensional variety $V$, a cycle $T$ in $V$, and a basis $\sigma_i$ for $H_{2d-1}(V)$, where $\sigma_i$ are cycles “draped over the saddles". We wish to find the coefficients $n_i$ in $T=\sum_{i}n_i \sigma_i$. If $\sigma_i^{*}$ is the dual basis, then $n_i=\sigma^{*}(T)$.

          Problem 2.2.

          [Robin Pemantle] If $d=2$, then we can find explicit $\sigma^*_i$. Can we do that if $d>2$?
            • Problem 2.3.

              [Robin Pemantle] Can we find $\sigma_i^*$ in de Rham cohomology?

                  Cite this as: AimPL: Analytic combinatorics in several variables, available at http://aimpl.org/combinseveral.