2021/02/19 by Nicholas Anderson, Anderson, Nicholas, Felipe Rincón +1
Computer Science · Medicine · #05B35 #14T10 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #Magnolia and Illicium research #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2102.09848
openalex publication_date 2021/02/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Tropical ideals are a class of ideals in the tropical polynomial semiring that combinatorially abstracts the possible collections of supports of all polynomials in an ideal over a field. We study zero-dimensional tropical ideals I with Boolean coefficients in which all underlying matroids are paving matroids, or equivalently, in which all polynomials of minimal support have support of size deg(I) or deg(I)+1 -- we call them paving tropical ideals. We show that paving tropical ideals of degree d+1 are in bijection with \mathbb Zn-invariant d-partitions of \mathbb Zn. This implies that zero-dimensional tropical ideals of degree 3 with Boolean coefficients are in bijection with \mathbb Zn-invariant 2-partitions of quotient groups of the form \mathbb Zn/L. We provide several applications of these techniques, including a construction of uncountably many zero-dimensional degree-3 tropical ideals in one variable with Boolean coefficients, and new examples of non-realizable zero-dimensional tropical ideals.