A Nullstellensatz for amoebas

Abstract

The amoeba of an affine algebraic variety $V\subset ({\mathbb{C}}^{*}){}^r$ is the image of $V$ under the map $(z_1,\dots ,z_r)\to (\log |z_1|,\dots ,\log |z_r|)$. We give a characterisation of the amoeba, based on the triangle inequality, which we call testing for lopsidedness. We show that if a point is outside the amoeba of $V$, there is an element of the defining ideal which witnesses this fact by being lopsided. This condition is necessary and sufficient for amoebas of arbitrary codimension as well as for compactifications of amoebas inside any toric variety. Our approach naturally leads to methods for approximating hypersurface amoebas and their spines by systems of linear inequalities. Finally, we remark that our main result can be seen as a precise analogue of a Nullstellensatz statement for tropical varieties