integralBasis -- Compute an integral basis of an algebraic function field.

Synopsis

Usage:

integralBasis(I)
Inputs:
- I, an ideal, in a PolynomialRing R of two variables over QQ, say x,y. defining an affine plane curve which has no singularities at infinity and its projective closure does not contain the point (x:y:z)=(0:1:0).
Outputs:
- a matrix, over frac(R)

Description

Computes an integral basis in CC(x)[y] of the integral closure of CC[x] in CC(x,y). We consider x as transcendental and y as algebraic. The i-th element of the integral basis has degree i as a polynomial in y. Note that the integral basis will have coefficients in QQ.

i1 : R=QQ[x,y]

o1 = R

o1 : PolynomialRing

i2 : I=ideal(y^8-x^3*(1+x)^5)

              8    8     7      6      5     4    3
o2 = ideal(- x  + y  - 5x  - 10x  - 10x  - 5x  - x )

o2 : Ideal of R

i3 : integralBasis(I)

o3 = | 1 x (x2+x)/y (x3+2x2+x)/y2 (x4+2x3+x2)/y3 (x5+3x4+3x3+x2)/y4
     ------------------------------------------------------------------------
     (x6+4x5+6x4+4x3+x2)/y5 (x7+4x6+6x5+4x4+x3)/y6 |

                    1              8
o3 : Matrix (frac R)  <--- (frac R)

Ways to use integralBasis :

integralBasis(Ideal)

For the programmer

The object integralBasis is a method function.