Abstract
We introduce the notion of tropical defects, certificates that a system of polynomial equations is not a tropical basis, and provide two algorithms for finding them in affine spaces of complementary dimension to the zero set. We use these techniques to solve open problems regarding del Pezzo surfaces of degree 3 and realizability of valuated gaussoids on 4 elements.
Highlights
The tropical variety Trop(I ) of a polynomial ideal I is the image of its algebraic variety under component-wise valuation
Inside mathematics for example, they enable new insights into important invariants in algebraic geometry [23] or the complexity of central algorithms in linear optimization [1]. They arise as spaces of phylogenetic trees in biology [25,29], loci of indifference prizes in
As the image of an algebraic variety, a tropical variety equals the intersection of all tropical hypersurfaces of the polynomials inside the ideal
Summary
The tropical variety Trop(I ) of a polynomial ideal I is the image of its algebraic variety under component-wise valuation. Inside mathematics for example, they enable new insights into important invariants in algebraic geometry [23] or the complexity of central algorithms in linear optimization [1] Outside mathematics, they arise as spaces of phylogenetic trees in biology [25,29], loci of indifference prizes in. We call Trop(F) a tropical prevariety and, if equality holds, F a tropical basis. This question is important for two main reasons. Current algorithms for computing tropical varieties require a Gröbner basis for each maximal Gröbner polyhedron, of which there can be many even for tropicalization of linear spaces [19].
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