Intersections of Amoebas

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-$1$.