Given the ideals I0,...,Ir in a polynomial ring R and the tuple a = (a0,...,ar) ∈ℕr+1 such that I0 is primary to the maximal homogeneous ideal of R, I1,...,Ir have positive height and a0+...+ar = dim R -1, the command computes the mixed multiplicity ea 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 ⊕I0u0I1u1...Irur/I0u0+1I1u1...Irur to calculate the mixed multiplicity. This setup enforces a0 ≠0. Due to the same reason, to compute the (a0+1, a1,..., ar)-th mixed multiplicity, one needs to enter the sequence a0,a1,...,ar 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 |