On the number of intersection points of the contour of an amoeba with a line

TL;DR

This paper studies the maximum number of intersection points between lines and amoeba contours of hypersurfaces in real space, introducing the concept of the -degree and providing bounds for it and related sets.

Contribution

It defines the -degree for amoeba contours and related sets, and establishes bounds for these degrees in the context of real hypersurface amoebas.

Findings

Bounds for the -degree of amoeba contours

Bounds for the -degree of amoeba boundaries

Bounds for the -degree of amoebas of real hypersurfaces

Abstract

In this note, we investigate the maximal number of intersection points of a line with the contour of hypersurface amoebas in $\mathbb{R}^n$. We define the latter number to be the $\mathbb{R}$-degree of the contour. We also investigate the $\mathbb{R}$-degree of related sets such as the boundary of amoebas and the amoeba of the real part of hypersurfaces defined over $\mathbb{R}$. For all these objects, we provide bounds for the respective $\mathbb{R}$-degrees.