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MixedMultiplicity :: mixedMultiplicity

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

Synopsis

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 :

For the programmer

The object mixedMultiplicity is a method function with options.