# 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 .