Some Results on $\mathrm{v}$-Number of Monomial Ideals
Liuqing Yang, Kaiwen Hu, Lizhong Chu
TL;DR
This work systematically analyzes the v-number for monomial ideals, linking it to reg(S/I) and establishing tight formulas for mixed product and edge ideals. It proves the conjectured behavior v(I^k) = α(I)k − 1 in broad cases where the base ideal has a linear resolution and no embedded primes, and extends the study to powers, square-free powers, and symbolic powers, deriving both general bounds and exact values in key graph families. The results illuminate how v-numbers evolve under taking powers and highlight connections to graph-theoretic invariants, offering concrete formulas for edge and complete graphs and proposing conjectures for line-graph powers. Collectively, the paper advances a coherent picture of how v-number interacts with regularity, powers of ideals, and combinatorial structures, with potential implications for coding-theoretic interpretations and invariant computations in combinatorial commutative algebra.
Abstract
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 $\mathrm{v}(I) \leq \mathrm{reg}(S/I)$. Next, we investigate an open conjecture on the v-number: if a monomial ideal $I$ has linear powers, then for all $k \geq 1$, $\mathrm{v}(I^k) = α(I)k - 1.$ We prove that if a monomial ideal $I$ with linear powers is a homogeneous square-free ideal and ($k \geq 1$) has no embedded associated primes, then $\mathrm{v}(I^k) = α(I)k - 1.$ 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.