2005/07/27 by Tristram Bogart, Anders Jensen, David Speyer +4 · 3 citations
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Basis (linear algebra) #Codimension #Combinatorics #Commutative Algebra and Its Applications #Discrete mathematics #Geometry #Ideal (ethics) #Mathematics #Polynomial and algebraic computation #Prime (order theory) #Pure mathematics #Tropical geometry #Variety (cybernetics) #math.AG #math.CO #msc:05E02 #msc:14Q04 #msc:68W04
paper · pdf · doi:10.1016/j.jsc.2006.02.004
published as J. Symb. Comput. 42 (2007), no. 1-2, 54--73 · 22 pages, 2 figures
arxiv created 2005/07/27 · openalex publication_date 2006/09/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
The tropical variety of a d-dimensional prime ideal in a polynomial ring with complex coefficients is a pure d-dimensional polyhedral fan. This fan is shown to be connected in codimension one. We present algorithmic tools for computing the tropical variety, and we discuss our implementation of these tools in the Gröbner fan software Gfan. Every ideal is shown to have a finite tropical basis, and a sharp lower bound is given for the size of a tropical basis for an ideal of linear forms.