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Amoebas of maximal area. (English) Zbl 0994.14032
Let \(f\in\mathbb{C} [X,Y]\) be a polynomial, \(\mathbb{C}^*= \mathbb{C}-\{0\}\), \(X=f^{-1}(0) \cap(\mathbb{C}^*)^2\) and \(\Delta\) the Newton polygon of \(f\). I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky introduced in their book “Discriminants, resultants and multidimensional determinants” (Boston 1994; Zbl 0827.14036) the notion of the amoeba \(A\) of \(f\), defined as the image \(A=\text{Log}(X)\) of \(X\) under the map \(\text{Log}: (\mathbb{C}^*)^2 \to\mathbb{R}^2: (z,w)\to(\text{Log}|z|,\text{Log}|w|)\). They also proved that the Lebesgue area Area\((A)\) is well defined. Later on, M. Passare and H. Rullgård showed [“Amoebas, Monge-Ampère measures and triangulations of the Newton polytope” (preprint; http://www.matematik.su.se/reports/2000/10)] the fundamental inequality \[ \text{Area}(A)\leq \pi^2\text{Area}(\Delta). \] The main result of the paper under review is a nice characterization of those \(f\)’s for which the above inequality is an equality. To explain it, the authors introduce some definitions. It is said that \(X\) is real up to multiplication by a constant if there exist \(a,b, c\in\mathbb{C}^*\) such that the polynomial \(af(X/b,Y/c)\) has real coefficients. The announced characterization is stated as follows:
Theorem. The following conditions are equivalent:
1. The equality Area\((A)=\pi^2 \text{Area} (\Delta)\) holds.
2. The curve \(X\) is real up to multiplication by a constant and the fibers of the restriction of Log to \(X\) have at most two points.
3. The curve \(X\) is real up to multiplication by a constant and its real part is a simple Harnack curve.
The paper is nicely written and it combines very cleverly elementary arguments of measure theory and real and complex algebraic geometry.

14P15 Real-analytic and semi-analytic sets
14H45 Special algebraic curves and curves of low genus
32C05 Real-analytic manifolds, real-analytic spaces
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