# Singular

#### D.6.9.4 param

Procedure from library hnoether.lib (see hnoether_lib).

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

Assume:
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"]).

Return:
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.

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

Example:
 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 ... print(Hne[1]); ==> 0,x, ==> 0,-1 print(hne_extended[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: param(Hne); ==> // ** Warning: result is exact up to order 2 in x and 3 in y ! ==> _[1]=-t2 ==> _[2]=-t3 param(hne_extended); ==> // ** 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 param(hne_extended,0); ==> // ** 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; param(hne); ==> // 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