Let R be the multi-homogeneous polynomial ring R=k[x1,0,x1,1,...,x1,r1, x2,0,x2,1,...,x2,r2, ......, xm,0,xm,1,...,xm,rm] and I be the multi-homogeneous ideal I=(f0,f1,...,fs) where the polynomials fi’s have the same multi-degree. We compute the degree of the rational map F: ℙr1 ×ℙr2 ×... ×ℙrm →ℙs defined by
This method calls "satSpecialFiber(I, nsteps)" in order to obtain the saturated special fiber ring and then computes the degree of F from the multiplicity of the saturated special fiber ring.
i1 : R = QQ[x,y,u,v, Degrees => {{1,0}, {1,0}, {0,1}, {0,1}}]
o1 = R
o1 : PolynomialRing
|
i2 : I = ideal(x*u, y*u, y*v) -- a birational map o2 = ideal (x*u, y*u, y*v) o2 : Ideal of R |
i3 : degreeOfMapIter(I, 5) o3 = 1 |
i4 : I = ideal(x*u, y*v, x*v + y*u) -- a non birational map o4 = ideal (x*u, y*v, y*u + x*v) o4 : Ideal of R |
i5 : degreeOfMapIter(I, 5) o5 = 2 |
i6 : A = matrix{ {x^5*u, x^2*v^2},
{y^5*v, x^2*u^2},
{0, y^2*v^2}
};
3 2
o6 : Matrix R <--- R
|
i7 : I = minors(2, A) -- a non birational
7 3 2 5 3 5 2 2 7 3
o7 = ideal (x u - x y v , x y u*v , y v )
o7 : Ideal of R
|
i8 : degreeOfMapIter(I, 5) o8 = 10 |
It only gives the correct answer if nteps is big enough to attain all the generators of the saturated special fiber ring.