# 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:

## 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 .