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arXiv · 2609.29654

On the Analyticity of Amoeba Contours

Mounir Nisse
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Abstract

The contour of an amoeba records the critical values of the logarithmic map and forms a fundamental interface between complex algebraic geometry and real-analytic geometry. Building on the known semianalyticity of amoeba contours, we prove the stronger statement that the componentwise exponential image of the contour of any algebraic hypersurface in $(\mathbb C^*)^n$ is semialgebraic. For a smooth hypersurface, we establish a natural criterion guaranteeing that the contour is a closed real-analytic hypersurface: the logarithmic Gauss map is transverse to $\mathbb RP^{n-1}$ and the logarithmic map has maximal rank along its critical locus. We then show that neither hypothesis is necessary, since singular or ramified critical parametrizations may still have complete real-analytic images. Finally, we derive intrinsic restrictions imposed by analyticity. In particular, every irreducible contour component of dimension $n-1$ is generated, over a dense open subset of its regular locus, by a real-analytic critical stratum on which the logarithmic map has maximal rank. These results distinguish the analytic geometry of the contour from the singularities of its critical parametrization and answer the question of Lang, Shapiro, and Shustin concerning effective sufficient conditions for analyticity.

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Mounir Nisse. 2026-08-29. On the Analyticity of Amoeba Contours. https://arxiv.org/abs/2609.29654

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