degreeOfMapIter -- computes the degree of a rational map

Synopsis

Usage:

degreeOfMapIter(I, nsteps)
Inputs:
- I, an ideal, an ideal defining the map
- nsteps, an integer, the number of steps used for computing the saturated special fiber ring
Outputs:
- an integer, the degree of the corresponding rational map

Description

Let $R$ be the multi-homogeneous polynomial ring $R=k[x_{1,0},x_{1,1},...,x_{1,r_1}, x_{2,0},x_{2,1},...,x_{2,r_2}, ......, x_{m,0},x_{m,1},...,x_{m,r_m}]$ and $I$ be the multi-homogeneous ideal $I=(f_0,f_1,...,f_s)$ where the polynomials $f_i$'s have the same multi-degree. We compute the degree of the rational map $\mathbb{F}: \mathbb{P}^{r_1} \times \mathbb{P}^{r_2} \times ... \times \mathbb{P}^{r_m} \to \mathbb{P}^s$ defined by $$ (x_{1,0} : ... : x_{1,r_1}; ...... ;x_{m,0} : ... : x_{m,r_m}) \to (f_0(x_{1,0},...,x_{1,r_1}, ...... ,x_{m,0},...,x_{m,r_m}), ..... , f_0(x_{1,0},...,x_{1,r_1}, ...... ,x_{m,0},...,x_{m,r_m})). $$ This method calls "satSpecialFiber(I, nsteps)" in order to obtain the saturated special fiber ring and then computes the degree of $\mathbb{F}$ 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

Caveat

It only gives the correct answer if nteps is big enough to attain all the generators of the saturated special fiber ring.

Ways to use degreeOfMapIter :

degreeOfMapIter(Ideal,ZZ)

For the programmer

The object degreeOfMapIter is a method function.