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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 n 2 and d 1 our construction yields a non-realizable degree d hypersurface scheme in Pⁿ. Maclagan-Rinc\'on produced a non-realizable line in Pⁿ for each n, and for (d, n) = (1, 2) the two constructions agree. An appendix by Mundinger compares the Macaulay construction with another method for canonically extending ideals to tropical ideals.
Fink et al. (Sat,) studied this question.