2025/04/06 by Yang, Liuqing, Hu, Kaiwen, Chu, Lizhong
#Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.2504.04478
This paper investigates the v-number of various classes of monomial ideals. First, we considers the relationship between the v-number and the regularity of the mixed product ideal I, proving that v(I) ≤ reg(S/I). Next, we investigate an open conjecture on the v-number: if a monomial ideal I has linear powers, then for all k ≥ 1, v(Ik) = α(I)k - 1. We prove that if a monomial ideal I with linear powers is a homogeneous square-free ideal and (k ≥ 1) has no embedded associated primes, then v(Ik) = α(I)k - 1. We have also drawn some conclusions about the k-th power of the graph.Additionally, we calculate the v-number of various powers of edge ideals(including ordinary power ,square-free powers, symbolic powers). Finally, we propose a conjecture that the v-number of ordinary powers of line graph is equal to the v-number of square-free powers.