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 full Jacobian dual rank 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 : jacobianDualRank I o3 = 2 |
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 : jacobianDualRank I o5 = 0 |
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 : jacobianDualRank I o8 = 0 |
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 : jacobianDualRank I o10 = 1 |
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 3 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 + y - 3x z - x*y*z - x*z + 4x w + 4x*y*w + 2x*z*w -
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2 2 2 2 2 2 2 2 3 2
y*z*w - 4x*w , x z + 2x*y*z + y z - x w - 2x*z*w - y*z*w - y*w , y z + 3x*z + y*z + z - x*y*w - 4x*z*w - 5y*z*w - 3z w +
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2 3
2z*w + 4w )
o13 : Ideal of R
|
i14 : jacobianDualRank I o14 = 3 |
i15 : I = ideal(random(2, R), random(2, R), random(2, R), random(2, R)); -- a non birational o15 : Ideal of R |
i16 : jacobianDualRank I o16 = 0 |
The object jacobianDualRank is a method function.