Asymptotic behaviour of Vasconcelos invariants for products and powers of graded ideals

Let $R$ be a commutative Noetherian $\mathbb{N}$-graded ring. Let $N\subseteq M$ be finitely generated $\mathbb{Z}$-graded $R$-modules. Let $I_1,\dots ,I_r$ be non-zero proper homogeneous ideals of $R$. Denote ${\mathbf{I}}^{\underset{‾}{n}}:=I_1^{n_1}\cdots I_r^{n_r}$ for $\underset{‾}{n}=(n_1,\dots ,n_r)\in {\mathbb{N}}^r$. In this paper, we prove that the (local) Vasconcelos invariant of ${\mathbf{I}}^{\underset{‾}{n}}M\mathrm{/}{\mathbf{I}}^{\underset{‾}{n}}N$ is eventually the minimum of finitely many linear functions in $\underset{‾}{n}$. The same holds for $M\mathrm{/}{\mathbf{I}}^{\underset{‾}{n}}N$ under certain conditions. Some specific examples are provided, where these functions are not eventually linear in $\underset{‾}{n}$. However, when $R$ is a polynomial ring over a field, we show that the global Vasconcelos invariants of $R\mathrm{/}{\mathbf{I}}^{\underset{‾}{n}}$ and ${\mathbf{I}}^{\underset{‾}{n}}\mathrm{/}{\mathbf{I}}^{\underset{‾}{n}+\underset{‾}{1}}$ are, in fact, asymptotically linear in $\underset{‾}{n}$ with the leading coefficients given by the initial degrees of $I_1,\dots ,I_r$. The last result is surprising: It differs from the Castelnuovo-Mumford regularity, which is not always linear even over polynomial rings, as shown by Bruns-Conca.