Projective hypersurfaces in tropical scheme theory I: the Macaulay ideal

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-Rincón 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.