Simon conjecture and the \(\text{ v }\)-number of monomial ideals

Let \(I\subset S\) $I\subset S$ be a graded ideal of a standard graded polynomial ring S with coefficients in a field K, and let \({\text {v}}(I)\) $\text{v}(I)$ be the \({\text {v}}\) $\text{v}$ -number of I. In previous work, we showed that for any graded ideal \(I\subset S\) $I\subset S$ , then \({\text {v}}(I^k)=\alpha (I)k+b\) $\text{v}(I^k)=\alpha (I)k+b$ , for all \(k\gg 0\) $k\gg 0$ , where \(\alpha (I)\) $\alpha (I)$ is the initial degree of I and b is a suitable integer. In the present paper, using polarization, we extend Simon conjecture to any monomial ideal. As a consequence, if Simon conjecture holds, I is a monomial ideal generated in a single degree and all powers of I have linear quotients, then \(b\in \{-1,0\}\) $b\in \{-1,0\}$ . This fact suggests that if I is an equigenerated monomial ideal with linear powers, then \({\text {v}}(I^k)=\alpha (I)k-1\) $\text{v}(I^k)=\alpha (I)k-1$ , for all \(k\ge 1\) $k\ge 1$ . We verify this conjecture for monomial ideals with linear powers having \({\text {depth}}\,S/I=0\) $\text{depth}S/I=0$ , edge ideals with linear resolution, polymatroidal ideals, and Hibi ideals.