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D.5.5.6 param

Procedure from library hnoether.lib (see hnoether_lib).

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

Assume:
L is the output of develop(f), or of extdevelop(develop(f),n), or one entry of the output of reddevelop(f).

Return:
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 "real" parametrization will be two power series and p[1],p[2] are truncations of these series.
- if the optional parameter x 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).

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);
==>  Warning! You have defined too many variables!
==>  All variables except the first two will be ignored!
==> h(0) = 1
==> a(1,2) = -1
list hne_extended=extdevelop(hne,10);
==> a(1,4) = 2
==> a(1,6) = -4
==> a(1,8) = 8
==> a(1,10) = -16
//   compare the 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
See also: develop; extdevelop.


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