Work

Projective hypersurfaces in tropical scheme theory I: the Macaulay ideal

Year: 2025

Type: article

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

Source: Research in the Mathematical Sciences

DOI: 10.1007/s40687-025-00517-7

Authors: Alex Fink, Jeffrey Giansiracusa, Noah Giansiracusa, Joshua Mundinger

Institutions: Queen Mary University of London, Durham University, Bentley University, University of Wisconsin–Madison

Language: English

Topic: Polynomial and algebraic computation

Subfield: Computational Theory and Mathematics

Field: Computer Science

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $4,090