2026/08/02 by Manohar Kumar, Emiliano Liwski
Mathematics · #math.AC #msc:13F20 #msc:05E40 #msc:05E99 #msc:13D45
40 Pages. Comments are welcome!
arxiv created 2026/08/02 · arxiv updated 2026/08/04
In this paper, we introduce a new framework for computing certain localized v-numbers of a class of ideals called coordinate-saturated ideals, which includes certain classes of Lovász-Saks-Schrijver (LSS) ideals and (generalized) binomial edge ideals associated with graphs. For a forest graph G, we derive an explicit formula for the localized v-number of the LSS ideal LG^\mathbbK(d), denoted by v_\mathfrakp∅(G)(LG^\mathbbK(d)), for all d ≥ 2, where \mathbbK is an algebraically closed field. As a consequence, we prove that v(LG^\mathbbK(d)) ≤ reg(R/LG^\mathbbK(d)), where v(LG^\mathbbK(d)) and reg(R/LG^\mathbbK(d)) denote the v-number of LG^\mathbbK(d) and the Castelnuovo-Mumford regularity of R/LG^\mathbbK(d), respectively. Also, we give an upper bound for v_I_Kn(LGℝ(2)), where ℝ is field of real numbers. Furthermore, we provide combinatorial descriptions of the localized v-number of parity binomial edge ideals, denoted by v_\mathfrakp+(G)(IG), and, as an application, show that v(IG) ≤ reg(R/IG) for several classes of non-bipartite graphs. Finally, we prove that v(JGk)≤ reg(R/JGk) for all powers of binomial edge ideals of closed graphs G.