partialJDRs -- computes the partial Jacobian dual ranks

Synopsis

Usage:

partialJDRs(I)
Inputs:
- I, an ideal, an ideal defining the map
Outputs:
- a list, the partial Jacobian dual ranks

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 function computes the partial Jacobian dual ranks of $\mathbb{F}$ (see Notation 4.2 in Degree and birationality of multi-graded rational maps).

First, we compute some examples in the bigraded setting.

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 : partialJDRs I

o3 = {1, 1}

o3 : List

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 : partialJDRs I

o5 = {0, 0}

o5 : List

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 : partialJDRs I

o8 = {0, 0}

o8 : List

i9 : I = ideal(x*u^2, y*u^2, x*v^2) -- non birational map

               2     2     2
o9 = ideal (x*u , y*u , x*v )

o9 : Ideal of R

i10 : partialJDRs I

o10 = {1, 0}

o10 : List

Next, we test some rational maps over three projective spaces.

i11 : R = QQ[x,y,z,w]

o11 = R

o11 : PolynomialRing

i12 : A = matrix{ {x + y,  x, x},
                  {3*z - 4*w, y, z},
                  {w,  z, z + w},
                  {y - z,  w, x + y}
                };

              4      3
o12 : Matrix R  <-- R

i13 : I = minors(3, A) -- a birational map

                      2       2      2    2                 2   2       2  
o13 = ideal (x*y*z + y z - x*z  - y*z  + y w - 2x*z*w + 4x*w , x y + x*y  +
      -----------------------------------------------------------------------
       3     2               2     2                                  2   2 
      y  - 3x z - x*y*z - x*z  + 4x w + 4x*y*w + 2x*z*w - y*z*w - 4x*w , x z
      -----------------------------------------------------------------------
                  2     2                        2   2        2      2    3  
      + 2x*y*z + y z - x w - 2x*z*w - y*z*w - y*w , y z + 3x*z  + y*z  + z  -
      -----------------------------------------------------------------------
                                  2        2     3
      x*y*w - 4x*z*w - 5y*z*w - 3z w + 2z*w  + 4w )

o13 : Ideal of R

i14 : partialJDRs I

o14 = {3}

o14 : List

i15 : I = ideal(random(2, R), random(2, R), random(2, R), random(2, R)); -- a non birational

o15 : Ideal of R

i16 : partialJDRs I

o16 = {0}

o16 : List

Ways to use partialJDRs :

partialJDRs(Ideal)

For the programmer

The object partialJDRs is a method function.