The mixed-multiplicity bound for full ideals

Let $(R,{\mathfrak m})$ be a regular local ring of dimension $d$, and let $t_i$ be defined by

$$(n+1)\cdots(n+d-1)=\sum_{i=1}^{d}t_i n^{d-i}.$$

An ideal is ${\mathfrak m}$-full if it has the corresponding maximal-ideal fullness property; every integrally closed ideal of positive height is ${\mathfrak m}$-full. The mixed-multiplicity bound for full ideals. For every ${\mathfrak m}$-full (in particular, integrally closed) ${\mathfrak m}$-primary ideal $I$,

$$\sum_{i=1}^{d}t_i\operatorname{e}_i({\mathfrak m}\mid I)\leq(d-1)!\mu(I),$$

where $\mu(I)$ is the minimal number of generators of $I$. This conjecture seeks to strengthen the known Dao–Smirnov bound for the first mixed multiplicity. The source does not resolve it in general.