2024/05/25 by Alex Fink, Fink, Alex, Jeffrey Giansiracusa +5
Computer Science · Mathematics · #14T10 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2405.16338
openalex publication_date 2024/05/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.