Amoebas of genus at most one

Details Amoebas of genus at most one Amoebas of genus at most one

2011-08-11

arxiv.org/abs/1108.2456v1

Mathematics

Algebraic Geometry

25 pages, 4 figures

Scientific paper

The amoeba of a Laurent polynomial $f \in \C[z_1^{\pm 1},..., z_n^{\pm 1}]$ is the image of its zero set $\mathcal{V}(f)$ under the Log-map. Understanding the configuration space of amoebas (i.e., the decomposition of the space of all polynomials, say, with given support or Newton polytope, with regard to the existing complement components) is a widely open problem. In this paper we investigate the class of polynomials $f$ whose Newton polytope $\New(f)$ is a simplex and whose support $A$ contains exactly one point in the interior of $\New(f)$. Amoebas of polynomials in this class may have at most one bounded complement component. We provide various results on the configuration space of these amoebas. In particular, we give upper and lower bounds in terms of the coefficients of $f$ for the existence of this complement component and show that the upper bound becomes sharp under some extremal condition. We establish connections from our bounds to Purbhoo's lopsidedness criterion and to the theory of $A$-discriminants. Finally, we provide a complete classification of the configuration space for the case that the exponent of the inner monomial is the barycenter of the simplex Newton polytope. In particular, we prove that the set of all polynomials with amoebas of genus 1 is path-connected in the corresponding configuration space, which proves a special case of the question on connectivity (for general Newton polytopes) stated by H. Rullg{\aa}rd.

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