Singular

D.4.12.1 integralBasis

Usage:

integralBasis(f, intVar); f irreducible polynomial in two variables, intVar integer indicating that the intVar-th variable of the ring is the integral element.
The base ring must be a ring in two variables, and the polynomial f must be monic as polynomial in the intVar-th variable.
Optional parameters in list choose (can be entered in any order):
Strategy:
- "global" -> computes the integral basis by global algorithms. This forces "normal" option.
- "local" -> computes the integral basis by computing the local contribution at each component of the singular
locus of R/<f>, and then putting everything together. (Default option.)
Algorithm:
- "normal" -> the integral bases are computed using the normalization algorithm.
- "hensel" -> the integral bases are computed using a special algorithm, based on Hensel lifting. (Default option.)
- "modular" -> uses modular algorithms for computing Groebner bases, radicals and decompositions whenever possible. Can be used together with any of the other options. The ground field must have characteristic 0. (Default option for ground fields of characteristic 0.)
- "nonModular" -> do not uses modular algorithms. (Default option for ground fields of positive charecteristic.)
- "rotation" -> apply a rotation when there are singularities with the same X-coordinate
- "noRotation" -> does not apply a rotation when there are singularities with the same X-coordinate (Default option.)
Other options:
- "atOrigin" -> will compute the local contribution at the origin to the integral basis, assuming that the curve has a singularity at the origin.
- "isIrred" -> assumes that the input polynomial f is irreducible, and therefore will not check this. If this option is given but f is not irreducible, the output might be wrong.
- list("inputJ", ideal inputJ) -> takes as initial test ideal the ideal inputJ. This option is only for use in other procedures. Using this option, the result might not be the integral basis. (When this option is given, the global option will be used.)
- list("inputC", ideal inputC) -> takes as initial conductor the ideal inputC. This option is only for use in other procedures. Using this option, the result might not be the integral basis. (When this option is given, the global option will be used.)
- "locBasis" -> when computing the integral basis at a prime or primary component, it computes a local basis, that is, a basis that is integral only over the ring localized at the component. This option is only valid when "atOrigin" is chosen or an initial test ideal or conductor is given.

Return:

a list, say l, of size 2.
l[1] is an ideal I and l[2] is a polynomial D such that the integral basis is b_0 = I[1] / D, b_1 = I[2] / D, ..., b_{n-1} = I[n] / D.
That is, the integral closure of k[x] in the algebraic function field k(x,y) is
k[x] b_0 + k[x] b_1 + ... + k[x] b_{n-1},
where we assume that x is the transcendental variable, y is the integral element (indicated by intVar), f gives the integral equation and n is the degree of f as a polynomial in y.

Theory:

We compute the integral basis of the integral closure of k[x] in k(x,y) by computing the normalization of the affine ring k[x,y]/<f> and converting the k[x,y]-module generators into a k[x]-basis.

LIB "integralbasis.lib";
printlevel = printlevel+1;
ring s = 0,(x,y),dp;
poly f = y5-y4x+4y2x2-x4;
list l = integralBasis(f, 2);
==> Computing the integral basis...
==> --Computing the associated primes of the singular locus...
==>   (Using non-modular algorithm.)
==> --Computing the integral basis at each component...
==> ----Computing the integral basis of component 
==> 1
==> ----Component: 
==> compo[1]=y
==> compo[2]=x
==> Integral basis computation finished.
l;
==> [1]:
==>    _[1]=x3
==>    _[2]=x3y
==>    _[3]=x2y2
==>    _[4]=xy3
==>    _[5]=y4+4x2y
==> [2]:
==>    x3
// The integral basis of the integral closure of Q[x] in Q(x,y) consists
// of the elements of l[1] divided by the polynomial l[2].
printlevel = printlevel-1;