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D.5.5.4 extdevelop

Procedure from library hnoether.lib (see hnoether_lib).

Usage:
extdevelop(L,N); list L, int N

Assume:
L is the output of develop(f), or of extdevelop(l,n), or one entry in the list hne in the ring created by hnexpansion(f[,"ess"]).

Return:
an extension of the Hamburger-Noether development of f as a list in the same format as L has (up to the last entry in the output of develop(f)).
Type help develop;, resp. help hnexpansion; for more details.

Note:
The new HN-matrix will have at least N columns (if the HNE is not finite). In particular, if f is irreducible then (in most cases) extdevelop(develop(f),N) will produce the same result as develop(f,N).
If the matrix M of L has n columns then, compared with parametrisation(L), paramametrize(extdevelop(L,N)) will increase the exactness by at least (N-n) more significant monomials.

Example:
 
LIB "hnoether.lib";
if (defined(HNEring))
{
def save_r_i_n_g=HNEring;
kill HNEring;
}
// ------ the example starts here -------
ring exring=0,(x,y),dp;
list hn=hnexpansion(x14-3y2x11-y3x10-y2x9+3y4x8+y5x7+3y4x6+x5*(-y6+y5)
-3y6x3-y7x2+y8);
def HNEring=hn[1];
setring HNEring;  
export(HNEring);  
==> // ** `HNEring` is already global
print(hne[1][1]);    // HNE of 1st branch is finite
==> 0,x,0,
==> 0,1,x 
print(extdevelop(hne[1],5)[1]);
==>  No extension is possible
==> 0,x,0,
==> 0,1,x 
print(hne[2][1]);    // HNE of 2nd branch can be extended
==> 0,x,0,
==> 0,1,x,
==> 0,1,-1
list ehne=extdevelop(hne[2],5);
print(ehne[1]);      // new HN-matrix has 5 columns
==> 0,x,0, 0,0,
==> 0,1,x, 0,0,
==> 0,1,-1,1,-1
parametrisation(hne[2]);
==> [1]:
==>    _[1]=x4-x5-x6+x7
==>    _[2]=x6-2x7+2x9-x10
parametrisation(ehne);
==> [1]:
==>    _[1]=x4-x5+x6-x7-x8+x9-x10+x11
==>    _[2]=x6-2x7+3x8-4x9+2x10-2x12+4x13-3x14+2x15-x16
if (defined(save_r_i_n_g))
{
kill HNEring;
def HNEring=save_r_i_n_g;
}
See also: develop; hnexpansion; parametrisation.


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