Projective hypersurfaces in tropical scheme theory I: the Macaulay ideal

TL;DR

This paper introduces the Macaulay tropical ideal, a canonical extension of principal ideals to tropical ideals with a universal property, revealing non-realizable tropical hypersurfaces in projective space.

Contribution

It presents a new construction based on transversal matroids that extends principal ideals to tropical ideals with a universal property, and explores non-realizable tropical hypersurfaces.

Findings

Constructs non-realizable degree d hypersurfaces in P^n for n≥2, d≥1.

Shows the Macaulay tropical ideal has a universal property among tropical ideal extensions.

Identifies cases where the construction coincides with known non-realizable tropical lines.

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 $n\geq 2$ and $d\geq 1$ our construction yields a non-realizable degree $d$ hypersurface scheme in $\mathbb{P}^n$. Maclagan-Rinc\'on 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.