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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.

MSC:
 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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