Amoebas of genus at most one

The amoeba of a Laurent polynomial $f \in \C[z_1^{\pm 1},\ldots,z_n^{\pm 1}]$ is the image of its zero set $\mathcal{V}(f)$ under the log-absolute-value map. Understanding the 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 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 space of amoebas for the case that the exponent of the inner monomial is the barycenter of the simplex Newton polytope. In particular, we show that the set of all polynomials with amoebas of genus 1 is path-connected in the corresponding space of amoebas, which proves a special case of the question on connectivity (for general Newton polytopes) stated by H. Rullg{\aa}rd.