Degree of 1-dimensional tropical prevariety

2024/04/09 by Grigoriev, Dima
#14T10 #Algebraic Geometry (math.AG) #FOS: Mathematics #G.2.1

paper · doi:10.48550/arxiv.2404.06440

Abstract

For a tropical prevariety V⊂ \RRⁿ (being a finite union of rational polyhedra) we define a tropical Hilbert function TH_V(k) to be the maximal number of tropically independent on V among tropical monomials with degrees at most k. In case dim V=1 we define the tropical degree as degT (V):=lim_{k→ ∞} (TH_V(k))/(k) \noindent and prove existence of the limit. We calculate explicitly (a modification of) the tropical degree of V when V=∪_l V_l is a star, i.e. the union of rays V_l with a common apex.

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• A tropical version of Hilbert polynomial (in dimension one)
• Tropical Tevelev degrees
• Equality of tropical rank and dimension for tropical linear series
• Upper bounds on Betti numbers of tropical prevarieties
• Tropical Convex Hull Computations