# mixedMultiplicity -- Compute a given mixed multiplicity of ideals in a polynomial ring.

## Synopsis

• Usage:
mixedMultiplicity (W1, W2)
• Inputs:
• W1, , sequence of ideals $I_0,...,I_r$ over a polynomial ring.
• W2, , tuple $a=(a_0,...,a_r)$ at which the mixed multiplicity is calculated
• Optional inputs:
• BasisElementLimit => ..., default value infinity, Bound the number of Groebner basis elements to compute in the saturation step
• DegreeLimit => ..., default value {}, Bound the degrees considered in the saturation step.
• MinimalGenerators => ..., default value true, Whether the saturation step returns minimal generators
• PairLimit => ..., default value infinity, Bound the number of s-pairs considered in the saturation step
• Strategy => ..., default value null, Choose a strategy for the saturation step
• Outputs:
• an integer, mixed multiplicity $e_a$ of ideals $I_0,...,I_r$.

## Description

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

 i1 : R = QQ[x,y,z,w] o1 = R o1 : PolynomialRing i2 : I = ideal(x*y*w^3,x^2*y*w^2,x*y^3*w,x*y*z^3) 3 2 2 3 3 o2 = ideal (x*y*w , x y*w , x*y w, x*y*z ) 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 $\oplus I_0^{u_0}I_1^{u_1}...I_r^{u_r}/I_0^{u_0+1}I_1^{u_1}...I_r^{u_r}$ to calculate the mixed multiplicity. This setup enforces $a_0 \neq 0.$ Due to the same reason, to compute the $(a_0+1, a_1,..., a_r)$-th mixed multiplicity, one needs to enter the sequence ${a_0,a_1,...,a_r}$ in the function. The same is illustrated in the following example.

 i5 : R = QQ[x,y,z] o5 = R o5 : PolynomialRing i6 : m = ideal vars R o6 = ideal (x, y, z) o6 : Ideal of R i7 : f = z^5 + x*y^7 + x^15 15 7 5 o7 = x + x*y + z o7 : R 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

## Ways to use mixedMultiplicity :

• mixedMultiplicity(Sequence,Sequence) (missing documentation)

## For the programmer

The object mixedMultiplicity is .