On the order map for hypersurface coamoebas

Abstract

In memory of Mikael Passare, who continues to inspire.Abstract. Given a hypersurface coamoeba of a Laurent polynomial f , it is an open problem to describe the structure of the set of connected components of its complement. In this paper we approach this problem by introducing the lopsided coamoeba. We show that the closed lopsided coamoeba comes naturally equipped with an order map, i.e. a map from the set of connected components of its complement to a translated lattice inside the zonotope of a Gale dual of the point configuration supp(f ). Under a natural assumption, this map is a bijection. Finally we use this map to obtain new results concerning coamoebas of polynomials of small codimension. 1

Imaginary projections of polynomials

“…For coamoebas, investigations are much more recent [11,12,22]. A prominent result states that the complement of an amoeba as well as the complement of the closure of a coamoeba consists of finitely many convex components, see [12,13]. As a key result, which also motivates our study, we show that the closure of the complement of the imaginary projection of a polynomial consists of finitely many convex components as well, see Theorem 4.1.…”

Feynman integrals as A-hypergeometric functions

“…These cycles are particularly useful in generic cases where polynomials may vanish on the integration region. In many cases, coamoebas are difficult to study analytically and one has to consider a rough version, which is called the lopsided coamoeba [51][52][53][54]. The relation between A-hypergeometric functions and coamoebas has been studied in Refs.…”

Coamoebas of Polynomials Supported on Circuits

“…As an edge Γ is one dimensional, the shell H f is a hyperplane arrangement. Its importance can be seen in that each full-dimensional cell of H f contain at most one connected component of the complement of C f , see [7].…”

“…The coamoeba of f (z) = 1 + z 1 + z 2 , as described in [7] and [14], can be seen in Figure 1, where it is drawn in the fundamental domains [−π, π] 2 and [0, 2π] 2 . The shell H f consist of the hyperplane arrangement drawn in black.…”

“…Acting on A by an integer affine transformation is equivalent to performing a monomial change of variables and multiplying f by a Laurent monomial. Such an action induces a linear transformation of the coamoeba C f , when viewed in R n [7]. We will repeatedly use this fact to impose assumptions on A, e.g., that it contains the origin.…”