Intersections of Amoebas

TL;DR

This paper explores the intersection properties of amoebas of multiple hypersurfaces, extending classical theorems and introducing a generalized order map to better understand their structure.

Contribution

It provides amoeba analogs of Bernstein's and Bézout's theorems for non-hypersurface varieties and generalizes the order map concept.

Findings

Upper bounds for the number of connected components of amoeba intersections

Extension of the order map to intersections of amoebas

The order map remains injective on each connected component

Abstract

Amoebas are projections of complex algebraic varieties in the algebraic torus under a Log-absolute value map, which have connections to various mathematical subjects. While amoebas of hypersurfaces have been intensively studied in recent years, the non-hypersurface case is barely understood so far. We investigate intersections of amoebas of $n$ hypersurfaces in $(\mathbb{C}^*)^n$, which are canonical supersets of amoebas given by non-hypersurface varieties. Our main results are amoeba analogs of Bernstein's Theorem and B\'ezout's Theorem providing an upper bound for the number of connected components of such intersections. Moreover, we show that the \emph{order map} for hypersurface amoebas can be generalized in a natural way to intersections of amoebas. In particular, analogous to the case of amoebas of hypersurfaces, the restriction of this generalized order map to a single connected component is still $1$-to-$1$.