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Projective hypersurfaces in tropical scheme theory I: the Macaulay ideal
Alex Fink1, Jeffrey Giansiracusa2, Noah Giansiracusa3
1Queen Mary University of London, London, England, UK.
Abstract:
A "tropical ideal" is an ideal in the idempotent semiring of tropical polynomials that is also, degree by degree, a tropical linear space. We introduce a construction based on transversal matroids that canonically extends any principal ideal to a tropical ideal. We call this the Macaulay tropical ideal. It has a universal property: any other extension of the given principal ideal to a tropical ideal with the expected Hilbert function is a weak image of the Macaulay tropical ideal. For each and , our construction yields a non-realizable degree d hypersurface scheme in . Maclagan-Rincón produced a non-realizable line in for each n, and for the two constructions agree. An appendix by Mundinger compares the Macaulay construction with another method for canonically extending ideals to tropical ideals.
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