2013/04/04 by Fu Liu, Liu, Fu · 1 citation
Computer Science · Mathematics · #05A15 #14N10 #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #Geometric and Algebraic Topology #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1304.1256
openalex publication_date 2013/04/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Based on results by Brugallé and Mikhalkin, Fomin and Mikhalkin give formulas for computing classical Severi degrees Nd, δ using long-edge graphs. In 2012, Block, Colley and Kennedy considered the logarithmic version of a special function associated to long-edge graphs appeared in Fomin-Mikhalkin's formula, and conjectured it to be linear. They have since proved their conjecture. At the same time, motivated by their conjecture, we consider a special multivariate function associated to long-edge graphs that generalizes their function. The main result of this paper is that the multivariate function we define is always linear. A special case of our result gives an independent proof of Block-Colley-Kennedy's conjecture. The first application of our linearity result is that by applying it to classical Severi degrees, we recover quadraticity of Qd, δ and a bound δ for the threshold of polynomiality of Nd, δ. Next, in joint work with Osserman, we apply the linearity result to a special family of toric surfaces and obtain universal polynomial results having connections to the Göttsche-Yau-Zaslow formula. As a result, we provide combinatorial formulas for the two unidentified power series B1(q) and B2(q) appearing in the Göttsche-Yau-Zaslow formula. The proof of our linearity result is completely combinatorial. We define τ-graphs which generalize long-edge graphs, and a closely related family of combinatorial objects we call (τ, n)-words. By introducing height functions and a concept of irreducibility, we describe ways to decompose certain families of (τ, n)-words into irreducible words, which leads to the desired results.