# mixedMultiplicity -- Compute a given mixed multiplicity of ideals

## Synopsis

• Usage:
mixedMultiplicity (W1, W2)
• Inputs:
• W1, , of ideals $I_0,\ldots,I_r$
• W2, , $a=(a_0,\ldots,a_r)$ at which the mixed multiplicity is calculated
• Outputs:
• an integer, mixed multiplicity $e_a$ of ideals $I_0,\ldots,I_r$

## Description

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.

 i1 : R = QQ[x,y,z,w]; i2 : I = ideal(x*y*w^3,x^2*y*w^2,x*y^3*w,x*y*z^3); o2 : Ideal of R i3 : m = ideal vars R; o3 : Ideal of R i4 : mixedMultiplicity ((m,I,I,I),(0,1,1,1)) o4 = 6

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.

 i5 : R = QQ[x,y,z]; i6 : m = ideal vars R; o6 : Ideal of R i7 : f = z^5 + x*y^7 + x^15; i8 : I = ideal(apply(0..2, i -> diff(R_i,f))) 14 7 6 4 o8 = ideal (15x + y , 7x*y , 5z ) o8 : Ideal of R i9 : mixedMultiplicity ((m,I),(2,0)) o9 = 1 i10 : mixedMultiplicity ((m,I),(1,1)) o10 = 4

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 $I = I_1 \cdots I_r.$

 i11 : S = QQ[x,y,z,w]/ideal(x*z, y*z); i12 : I = ideal(x,y); o12 : Ideal of S i13 : m = ideal vars S; o13 : Ideal of S i14 : K = saturate(sub(ideal(),S), I^2); o14 : Ideal of S i15 : T = S/K; i16 : J = sub(I, T); o16 : Ideal of T i17 : n = sub(m, T); o17 : Ideal of T i18 : mixedMultiplicity ((n,J,J),(2,0,0)) o18 = 1