Amoeba-shaped polyhedral complex of an algebraic hypersurface (Q2419738)

In the paper under review the authors study amoeba-shaped polyhedral complexes of algebraic hypersurfaces. The amoeba \(\mathcal{A}_{f}\) of a Laurent polynomial \(f(x)\) (or of the algebraic hypersurface \( \{ f(x) = 0\}\)) is defined to be the image of the hypersurface \(f^{-1}(0)\) under the map \[ \mathrm{Log}:(x_{1},\dots,x_{n})\rightarrow(\log|x_{1}|,\dots,\log|x_{n}|). \] For \(n > 1\), the amoeba of a polynomial is a closed unbounded semi-analytic subset of the real vector space \(\mathbb{R}^{n}\). The weighted moment map is associated with the algebraic hypersurface \(\{x\in\mathbb{C}^{n} : f(x) :=\sum_{s\in S} a_{s} x^{s} = 0\}\) through \[ \mu_{f}(x) = \frac{\sum_{s\in S} s\cdot |a_{s}||x^{s}|} {\sum_{s\in S} |a_{s}||x^{s}|}. \] By the weighted compactified amoeba of an algebraic hypersurface \(H = \{x\in \mathbb{C}^{n} : f (x) = 0\}\) we will mean the set \(\mu_{f}(H)\). We denote it by \(\mathcal{WCA}(f)\). Finally, the Hadamard power of order \(r \in \mathbb{R}\) of a polynomial \(f(x) = \sum_{s\in S}a_{s}x^{s}\) is defined to be \(f^{[r]}(x):= \sum_{s \in S}a_{s}^{r}x^{s}\). Main Theorem. Let \(f\) be a polynomial \(\mathbb{C}[x_{1}^{\pm 1},\dots,x_{n}^{\pm 1}]\) with the Newton polytope \(\mathcal{N}\) such that \(|a_{\alpha}| \geq 1\) for every \(\alpha \in\mathrm{Vert}(\mathcal{N})\). Assume that the function which assigns to each \(\alpha \in \mathcal{N} \cap \mathbb{Z}^{n}\) the real number \(\log |a_{\alpha}|\) is concave and the subdivision of \(\mathcal{N}\) dual to the tropical hypersurface \(\Gamma\) associated to the tropical polynomial \(f_{\mathrm{trop}}\) defined by \[ f_{\mathrm{trop}}(\xi)=\max_{\alpha\in\mathcal{N}\cap\mathbb{Z}^{n}}\{\log | a_{\alpha}| + \langle \alpha, \xi\rangle \} \] is a triangulation. Then the set-theoretical limit \[ \mathcal{P}^{\infty}_{f} :=\lim_{r\rightarrow \infty} \mathcal{WCA}(f^{[r]}) \] is a polyhedral complex. Moreover, its complement in \(\mathcal{N}\) has the same topology of the complement of the amoeba of \(\mathcal{A}\) determined by \(f\), i.e., \(\pi_{0}(\mathbb{R}^{n} \setminus \mathcal{A}) = \pi_{0}(\mathcal{N} \setminus \mathcal{P}^{\infty}_{f})\). In particular, if \(n=2\) then \(\mathcal{P}^{\infty}_{f}\) is a simplicial complex.