2024/03/30 by Manohar Kumar, Kumar, Manohar, Ramakrishna Nanduri +3 · 2 citations
Mathematics · #05E40 #13A02 #13F20 #13F55 #Advanced Optimization Algorithms Research #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2404.00493
openalex publication_date 2024/03/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we study the asymptotic behaviour of the v-number of a Noetherian graded filtration I= \I[k]\k≥ 0 of a Noetherian ℕ-graded domain R. Recently, it is shown that v(I[k]) is periodically linear in k for k ≫ 0. We show that all these linear functions have the same slope, i.e. limk → ∞\fracv(I[k])k exists, which is equal to limk → ∞\fracα(I[k])k, where α(I) denotes the minimum degree of a non-zero element in I. In particular, for any Noetherian symbolic filtration I= \I(k)\k≥ 0 of R, it follows that limk → ∞\fracv(I(k))k=α(I), the Waldschmidt constant of I. Next, for a non-equigenerated square-free monomial ideal I, we prove that v(I(k)) ≤ reg(R/I(k)) for k≫ 0. Also, for an ideal I having the symbolic strong persistence property, we give a linear upper bound on v(I(k)). As an application, we derive some criteria for a square-free monomial ideal I to satisfy v(I(k))≤ reg(R/I(k)) for all k≥ 1, and provide several examples in support. In addition, for any simple graph G, we establish that v(J(G)(k)) ≤ reg(R/J(G)(k)) for all k ≥ 1, and v(J(G)(k)) = reg(R/J(G)(k))=α(J(G)(k))-1 for all k≥ 1 if and only if G is a Cohen-Macaulay very-well covered graph, where J(G) is the cover ideal of G.