Abstract:

The coamoeba (or alga) of an algebraic hypersurface $X\subset\mathbf{C}_*^n$ is by definition the image of $X$ under the mapping $(z_1,\ldots,z_n)\mapsto(\arg z_1,\ldots,\arg z_n)$. We will consider the case where $X$ is a discriminant locus associated with an integer matrix $B$ of size $n\times 2$. We prove that the sum of the coamoeba $\text{Arg}(X)$ and the zonotope $\Pi_B$ built from the row vectors of $B$, both considered as simplicial chains, is equal to $d_B$ times the fundamental class of the torus $(\mathbf{R}/2\pi\mathbf{Z})^2$. Here $d_B$ is the degree of the dual toric variety $X^\vee$. It also coincides with the dimension of the space of solutions to the $A$-hypergeometric system, where $A$ denotes the Gale dual of $B$.

This is joint work with Lisa Nilsson.