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MultiGradedRationalMap :: degreeOfMapIter

degreeOfMapIter -- computes the degree of a rational map

Synopsis

Description

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

(x1,0 : ... : x1,r1; ...... ;xm,0 : ... : xm,rm) →(f0(x1,0,...,x1,r1, ...... ,xm,0,...,xm,rm), ..... , f0(x1,0,...,x1,r1, ...... ,xm,0,...,xm,rm)).

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

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 :