Projective hypersurfaces in tropical scheme theory I: the Macaulay ideal

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 $n\ge 2$ and $d\ge 1$ our construction yields a non-realizable degree $d$ hypersurface scheme in ${\mathbb{P}}^n$. Maclagan-Rincón produced a non-realizable line in ${\mathbb{P}}^n$ 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.

Appendix by Joshua Mundinger. 21pp