2016/07/25 by Kazunori Matsuda, Matsuda, Kazunori, Tao Suzuki +3 · 1 citation
Computer Science · Engineering · Mathematics · #13A02 #13A15 #13C15 #Advanced Numerical Analysis Techniques #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1607.07223
openalex publication_date 2016/07/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a nonincreasing function f : ℤ≥ 0 ∖ \ 0 \ → ℤ≥ 0 such that (i) f(k) - f(k+1) ≤ 1 for all k ≥ 1 and (ii) if a = f(1) and b = limk → ∞ f(k), then |f-1(a)| ≤ |f-1(a-1)| ≤ ⋯ ≤ |f-1(b+1)|, a system of generators of a monomial ideal I ⊂ K[x1, …, xn] for which \rm depth S/Ik = f(k) for all k ≥ 1 is explicitly described. Furthermore, we give a characterization of triplets of integers (n,d,r) with n > 0, d ≥ 0 and r > 0 with the properties that there exists a monomial ideal I ⊂ S = K[x1, …, xn] for which limk → ∞ \rm depth S/Ik = d and \rm dstab(I) = r, where \rm dstab(I) is the smallest integer k0 ≥ 1 with \rm depth S/Ik0 = \rm depth S/Ik0+1 = \rm depth S/Ik0+2 = ⋯.