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D.6.11.4 param

Procedure from library hnoether.lib (see hnoether_lib).

param(L [,s]); L list, s any type (optional)

L is the output of develop(f), or of extdevelop(develop(f),n), or (one entry in) the list of HN data created by hnexpansion(f[,"ess"]).

If L are the HN data of an irreducible plane curve singularity f: a parametrization for f in the following format:
- if only the list L is given, the result is an ideal of two polynomials p[1],p[2]: if the HNE was finite then f(p[1],p[2])=0}; if not, the true parametrization will be given by two power series, and p[1],p[2] are truncations of these series.
- if the optional parameter s is given, the result is a list l: l[1]=param(L) (ideal) and l[2]=intvec with two entries indicating the highest degree up to which the coefficients of the monomials in l[1] are exact (entry -1 means that the corresponding parametrization is exact).
If L collects the HN data of a reducible plane curve singularity f, the return value is a list of parametrizations in the respective format.

If the basering has only 2 variables, the first variable is chosen as indefinite. Otherwise, the 3rd variable is chosen.

LIB "hnoether.lib";
ring exring=0,(x,y,t),ds;
poly f=x3+2xy2+y2;
list Hne=develop(f);
list hne_extended=extdevelop(Hne,10);
//   compare the HNE matrices ...
==> 0,x,
==> 0,-1
==> 0,x, 0,0,0,0, 0,0,0,0, 
==> 0,-1,0,2,0,-4,0,8,0,-16
// ... and the resulting parametrizations:
==> // ** Warning: result is exact up to order 2 in x and 3 in y !
==> _[1]=-t2
==> _[2]=-t3
==> // ** Warning: result is exact up to order 10 in x and 11 in y !
==> _[1]=-t2+2t4-4t6+8t8-16t10
==> _[2]=-t3+2t5-4t7+8t9-16t11
==> // ** Warning: result is exact up to order 10 in x and 11 in y !
==> [1]:
==>    _[1]=-t2+2t4-4t6+8t8-16t10
==>    _[2]=-t3+2t5-4t7+8t9-16t11
==> [2]:
==>    10,11
// An example with more than one branch:
list L=hnexpansion(f*(x2+y4));
==> // 'hnexpansion' created a list of one ring.
==> // To see the ring and the data stored in the ring, type (if you assigned
==> // the name L to the list):
==>      show(L);
==> // To display the computed HN expansion, type
==>      def HNring = L[1]; setring HNring;  displayHNE(hne); 
def HNring = L[1]; setring HNring;
==> // Parametrization of branch number 1 computed.
==> // ** Warning: result is exact up to order 4 in x and 5 in y !
==> // Parametrization of branch number 2 computed.
==> // Parametrization of branch number 3 computed.
==> [1]:
==>    _[1]=-x2+2*x4
==>    _[2]=-x3+2*x5
==> [2]:
==>    _[1]=(a)*x2
==>    _[2]=x
==> [3]:
==>    _[1]=(-a)*x2
==>    _[2]=x
See also: develop; extdevelop.