Given the ideals $I_0,\ldots,I_r$ in a ring $R$ and the tuple $a = (a_0,\ldots,a_r) \in \mathbb{N}^{r+1}$ such that $I_0$ is primary to the maximal homogeneous ideal of $R$, $I_1,\ldots,I_r$ have positive grade and $a_0+ \cdots +a_r = dim R 1$, the function computes the mixed multiplicity $e_a$ of the ideals.




The function computes the Hilbert polynomial of the graded ring $\bigoplus (I_0^{u_0}I_1^{u_1} \cdots I_r^{u_r}/I_0^{u_0+1}I_1^{u_1} \cdots I_r^{u_r})$ to compute the mixed multiplicity. If the ideals $I_1,\ldots,I_r$ are also primary to the maximal ideal, then to compute the $(a_0+1, a_1,\ldots, a_r)$th mixed multiplicity, one needs to enter the sequence ${a_0,a_1,\ldots,a_r}$ in the function. The same is illustrated in the following example.






In case the user wants to compute a mixed multiplicity of ideals where one (or many) ideal(s) doesn't have positive grade, then one can pass to the ring $R/(0:I^\infty),$ where $(0:I^\infty)$ denotes the saturation of the ideal $I = I_1 \cdots I_r.$








The object mixedMultiplicity is a method function.