Depth and Stanley depth of powers of the path ideal of a cycle graph

2023/03/27 by Silviu Bălănescu, Balanescu, Silviu, Mircea Cimpoeaş +1 · 1 citation
Computer Science · Mathematics · Medicine · #13C15 #13F20 #13P10 #Cholinesterase and Neurodegenerative Diseases #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2303.15032

openalex publication_date 2023/03/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let J_n,m:=(x₁x₂⋯ x_m, x₂x₃⋯ x_m+1, …, x_n-m+1⋯ x_n, x_n-m+2⋯ x_nx₁, …, x_nx₁⋯ x_m-1) be the m-path ideal of the cycle graph of length n, in the ring S=K[x₁,…,x_n]. Let d=gcd(n,m). We prove that depth(S/J_n,m^t)≤ d-1 for all t≥ n-1. We show that sdepth(S/J_n,n-1^t)=depth(S/J_n,n-1^t)=max\n-t-1,0\ for all t≥ 1. Also, we give some bounds for depth(S/J_n,m^t) and sdepth(S/J_n,m^t), where t≥ 1.

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• Depth of the 3-path Ideal of Square of a Path

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