A Nullstellensatz for amoebas Export

@misc{citeulike:543539, abstract = {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.}, archivePrefix = {arXiv}, author = {Purbhoo, Kevin}, citeulike-article-id = {543539}, citeulike-linkout-0 = {http://arxiv.org/abs/math.AG/0603201}, citeulike-linkout-1 = {http://arxiv.org/pdf/math.AG/0603201}, day = {8}, eprint = {math.AG/0603201}, keywords = {algebraic-geometry, amoeba, nullstellensatz, tropical}, month = {Mar}, posted-at = {2006-03-10 06:55:26}, priority = {2}, title = {A Nullstellensatz for amoebas}, url = {http://arxiv.org/abs/math.AG/0603201}, year = {2006} }

TY - ELEC ID - citeulike:543539 L3 - citeulike-article-id:543539 N2 - 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. TI - A Nullstellensatz for amoebas KW - algebraic-geometry KW - amoeba KW - nullstellensatz KW - tropical AU - Purbhoo, Kevin PY - 2006/Mar/08/ UR - http://arxiv.org/abs/math.AG/0603201 ER -