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ICfractions -- Compute the fractions integral over a domain.

Synopsis

Usage:
ICfractions R
Inputs:
- R, a ring that is an affine domain
Outputs:
- a matrix whose entries are fractions that generate the integral closure of R over R.
Optional inputs:
- Strategy => ..., -- Allows the user to obtain all of the fractions considered in the process of building the integral closure

Description

i1 : R = QQ[x,y,z]/ideal(x^6-z^6-y^2*z^4);

i2 : ICfractions R

o2 = | x3/z2 x2/z x y z |

            /      QQ [x, y, z] \1      /      QQ [x, y, z] \5
o2 : Matrix |frac --------------|  <--- |frac --------------|
            |      6    2 4    6|       |      6    2 4    6|
            \     x  - y z  - z /       \     x  - y z  - z /

i3 : integralClosure(R,Variable => a)

                  QQ [w , w , x, y, z]
                       7   6
o3 = ---------------------------------------------
       2                    2         2    2    2
     (x  - w z, w x - w z, w  - w x, w  - y  - z )
            6    6     7    6    7    7

o3 : QuotientRing

Thus the new variables w_7 and w_6 correspond to the fractions respectively. The program currently also returns the original variables as part of the matrix. In this way the user can see if any are simplified out of the ring during the process of computing the integral closure.

The fractions returned correspond to the variables returned by the function integralClosure. The function integralClosure eliminates redundant fractions during its iteration. If the user would like to see all fractions generated during the computation, use the optional argument Strategy => Long as illustrated here.

i4 : ICfractions(R, Strategy => Long)

o4 = | x3/z2 x2/z (y2z+z3)/x x3/z x4/z2 (y2z2+z4)/x x4/z (y2z3+z5)/x x y z |

            /      QQ [x, y, z] \1      /      QQ [x, y, z] \11
o4 : Matrix |frac --------------|  <--- |frac --------------|
            |      6    2 4    6|       |      6    2 4    6|
            \     x  - y z  - z /       \     x  - y z  - z /

Ways to use `ICfractions` :

ICfractions(Ring)