Projective hypersurfaces in tropical scheme theory I: the Macaulay ideal

Fink, Alex; Giansiracusa, Jeffrey; Giansiracusa, Noah; Mundinger, Joshua

Abstract

A "tropical ideal" is an ideal in the idempotent semiring of tropical polyno-mials 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 P n. Maclagan-Rincón produced a non-realizable line in 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.

Citation

Fink, A., Giansiracusa, J., Giansiracusa, N., & Mundinger, J. (2025). Projective hypersurfaces in tropical scheme theory I: the Macaulay ideal. Research in the Mathematical Sciences, 12, Article 30. https://doi.org/10.1007/s40687-025-00517-7

• Journal Article Type: Article
• Acceptance Date: Mar 20, 2025
• Online Publication Date: Apr 25, 2025
• Publication Date: Apr 25, 2025
• Deposit Date: Mar 21, 2025
• Publicly Available Date: May 21, 2025
• Journal: Research in the Mathematical Sciences
• Print ISSN: 2522-0144
• Electronic ISSN: 2197-9847
• Publisher: Springer
• Peer Reviewed: Peer Reviewed
• Volume: 12
• Article Number: 30
• DOI: https://doi.org/10.1007/s40687-025-00517-7
• Keywords: Tropical geometry, scheme theory, tropical ideals
• Public URL: https://durham-repository.worktribe.com/output/3721195