A Nullstellensatz for amoebas

By Kevin Purbhoo. 2006

73 citations recorded by the source corpus.

The amoeba of an affine algebraic variety V in (C^*)^r is the image of V under the map (z_1, ..., z_r) -> (log|z_1|, ..., 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 a precise analogue of a Nullstellensatz statement for tropical varieties.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/899ad9e4cd6b11de732024875a68024ff0424c30

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In Algebra and Number Theory

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Citations in the corpus

Amoebas, nonnegative polynomials and sums of squares supported on circuits (2014, background): “Part (3) is an immediate consequence of Purbhoo’s lopsidedness condition (also referred as generalized Pellet’s Theorem), see [30].”
On the frontiers of polynomial computations in tropical geometry (2004, methodology): “N P-hardness of deciding whether an amoeba intersects a coordinate hyperplane.) For some Nullstellensatz-type algorithmic results see [ 19 ].”