2013/07/22 by Jimmy Jianyun Shan, Shan, Jimmy Jianyun
Mathematics · #05E40 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #math.AC #math.CO #msc:05E40
paper · pdf · doi:10.48550/arxiv.1307.5895
The new version proves the 3 variable case, but the general case is open. Comments and suggestions are welcome! arXiv admin note: text overlap with arXiv:math/0301153 by other authors
openalex publication_date 2013/07/22 · arxiv created 2014/02/14 · arxiv updated 2014/02/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a graph G, Postnikov-Shapiro \citePS04 construct two ideals IG and JG. IG is a monomial ideal and JG is generated by powers of linear forms. They proved the equality of their Hilbert series and conjectured that the graded Betti numbers are equal. When G=Kn+1l,k is the complete graph on the vertices \0,1,⋯, n\ with the edges ei, j, i, j≠ 0, of multiplicity k and the edges e0, i of multiplicity l, for two non-negative integers k and l, they gave an explicit formula for the graded Betti numbers of IG, which are conjecturally the same for JG. We prove this conjecture in the case n=3, which was also conjectured by Schenck \citeS04.