2015/09/18 by Drew Johnson, Johnson, Drew
Computer Science · Mathematics · #14M25 #14N10 #14T05 #52B20 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Mathematics and Applications #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1509.05815
openalex publication_date 2015/09/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the question of when points in tropical affine space uniquely determine a tropical hypersurface. We introduce a notion of multiplicity of points so that this question may be meaningful even if some of the points coincide. We give a geometric/combinatorial way and a tropical linear-algebraic way to approach this question. First, given a fixed hypersurface, we show how one can determine whether points on the hypersurface determine it by looking at a corresponding marking of the dual complex. With a regularity condition on the dual complex and when the number of points is minimal, we show that our condition is equivalent to the connectedness of an appropriate sub-complex. Second, we introduce notions of non-singularity of tropical matrices and solutions to tropical linear equations that take into account our notion of multiplicity and prove a Cramer's Rule type theorem relating them.