On the curvature of the real amoeba

Jean-Jacques Risler; Mikael Passare

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On the curvature of the real amoeba

Mikael Passare, Jean-Jacques Risler

Algebraic Geometry · Submitted 30 Dec 2010

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We describe an universal bound for the total curvature of the real amoeba $\mathcal{A}_{\mathbb{R} A}$ and we prove that this bound is reached if and only if the curve $A$ is a simple Harnack curve in the sense of Mikhalkin.

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What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.12 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.65 · compares with prior results 0.16 · mentions code or data availability 0.02.

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arXiv listing abstract v1. Results and the main claim are underlined; show every role.

For a real smooth algebraic curve $A \subset (\mathhbb{C}^*)^2$, the amoeba $\mathcal{A} \subset \mathbb{R}^2$ is the image of $A$ under the map Log : $(x,y) \mapsto (\log |x|, \log | y |)$. We describe an universal bound for the total curvature of the real amoeba $\mathcal{A}_{\mathbb{R} A}$ and we prove that this bound is reached if and only if the curve $A$ is a simple Harnack curve in the sense of Mikhalkin.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	For a real smooth algebraic curve $A \subset (\mathhbb{C}^*)^2$, the amoeba $\mathcal{A} \subset \mathbb{R}^2$ is the image of $A$ under the map Log : $(x,y) \mapsto (\log \|x\|, \log \| y \|)$.	background 1.00	0.07	0.02	asserted 0.99
s1	We describe an universal bound for the total curvature of the real amoeba $\mathcal{A}_{\mathbb{R} A}$ and we prove that this bound is reached if and only if the curve $A$ is a simple Harnack curve in the sense of Mikhalkin.	result 0.77	0.94	0.93	asserted 1.00

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Original abstract

For a real smooth algebraic curve $A \subset (\mathhbb{C}^*)^2$, the amoeba $\mathcal{A} \subset \mathbb{R}^2$ is the image of $A$ under the map Log : $(x,y) \mapsto (\log |x|, \log | y |)$. We describe an universal bound for the total curvature of the real amoeba $\mathcal{A}_{\mathbb{R} A}$ and we prove that this bound is reached if and only if the curve $A$ is a simple Harnack curve in the sense of Mikhalkin.

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