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<p>We investigate the v-number of various classes of monomial ideals. First, we consider the relationship between the v-number and the regularity of the mixed product ideal <i>I</i>, proving that v(<i>I</i>) ⩽ reg(<i>S/I</i>). Next, we investigate an open conjecture on the v-number: if a monomial ideal <i>I</i> has linear powers, then for all <i>k</i> ⩾ 1, v(<i>I</i><sup><i>k</i></sup>) = <i>α</i>(<i>I</i>)<i>k</i> − 1. We prove that if a monomial ideal <i>I</i> with linear powers and <i>I</i><sup><i>k</i></sup> (for any <i>k</i> ⩾ 1) has no embedded associated primes, then v(<i>I</i><sup><i>k</i></sup>) = <i>α</i>(<i>I</i>)<i>k</i> − 1. Additionally, we calculate the v-number of ordinary power and square-free power of edge ideal. 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.</p>

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Some results on v-number of monomial ideals

  • 期刊论文

  • 发表日期：

    2025年11月10日

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  • Liuqing Yang,
  • Kaiwen Hu,
  • Lizhong Chu

摘要

We investigate the v-number of various classes of monomial ideals. First, we consider 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 and Ik (for any k ⩾ 1) has no embedded associated primes, then v(Ik) = α(I)k − 1. Additionally, we calculate the v-number of ordinary power and square-free power of edge ideal. 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.