You get P1 and P2 with running primaryDecomposition (of the ideal). Then Ki = R/Pi.
The integral closure of R equals the integral closure of R/P1 \oplus R/P2.
Once you see the presentation of R/P1, the equation of integral dependence of z/y and z/v is easy by observation of the one relevant binomials in the generating set. Namely, one of the generators of P1 is y^2 v - z^2,
so an equation of integral dependence of z/y is (z/y)^2 - v. Since a Groebner basis of P1 also contains yv^5-z^5, then an equation of integral dependence of z/v is (z/v)^5 - y.
i1 : input "mac/intclMay2022"
ii2 : R = ZZ/7[x,y,z,v,h,f]
oo2 = R
oo2 : PolynomialRing
ii3 : I = ideal (y^2*v-4*x^9-z^2,x*h-z*v^2+y^5,x*f-v^3+y^3*z)
9 2 2 5 2 3 3
oo3 = ideal (3x + y v - z , y - z*v + x*h, y z - v + x*f)
oo3 : Ideal of R
ii4 : I == radical I -- true, intersection of two prime ideals, both of height 3
oo4 = true
ii5 : primaryDecomposition I
2 2 3 4 3 3 5 2 3 3
oo5 = {ideal (x, y v - z , y*z - v , y z - v , y - z*v ), ideal (y z - v +
--------------------------------------------------------------------------
2 3 2 2 2 5 2 3 2 6 2
x*f, y v - z v - x*y f + x*z*h, y - z*v + x*h, y*z v - v - x*y*z h +
--------------------------------------------------------------------------
3 2 2 9 2 2 8 2 2 8 3
2x*v f - x f , x - 2y v + 2z , x v - 2y f + 2z*h, x y + 2v*h - 2z*f)}
oo5 : List
ii6 :
i7 : icFracP(R/oo5_0)
2 2
z z y*z y v v
o7 = {1, -, -, ---, --, --, -}
y v v v z y
o7 : List