The v-function formula for powers of ideals generated by regular sequences

Let $S$ be the polynomial ring under consideration, and let $f_1,\dots,f_m$ be a homogeneous regular sequence on $S$. For each $i$, write

$$f_i=g_{i,1}^{a_{i,1}}\cdots g_{i,b_i}^{a_{i,b_i}}$$

for its decomposition into irreducible polynomials, and set $I=(f_1,\dots,f_m)$. Here, $\alpha(I)$ denotes the least degree of a nonzero homogeneous element of $I$, and $\v(I^k)$ is the $v$-function of $I^k$.

The v-function formula. For all $k\ge1$,

$$\v(I^k)=\alpha(I)k+\sum_{i=1}^m\left(\deg f_i-\max_{1\le j\le b_i}\deg g_{i,j}\right)-\alpha(I).$$

This is proposed in the context of formulas for the $v$-function of powers of sums of ideals. The surrounding results establish the formula for certain equigenerated vertex splittable ideals, while the statement here extends the expected behavior to ideals generated by homogeneous regular sequences; its resolution is not indicated in the supplied text.