#### D.6.15.3 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 `parametrization(L)`, `paramametrize(extdevelop(L,N))` will increase the exactness by at least (N-n) more significant monomials.

Example:

    LIB "hnoether.lib";
    ring exring=0,(x,y),dp;
    list Hne=hnexpansion(x14-3y2x11-y3x10-y2x9+3y4x8+y5x7+3y4x6+x5*(-y6+y5)
    -3y6x3-y7x2+y8);
    ==> // No change of ring necessary, return value is HN expansion.
    displayHNE(Hne);    // HNE of 1st,3rd branch is finite
    ==> // Hamburger-Noether development of branch nr.1:
    ==>   y = z(1)*x
    ==>   x = z(1)^2
    ==>
    ==> // Hamburger-Noether development of branch nr.2:
    ==>   y = z(1)*x
    ==>   x = z(1)^2+z(1)^2*z(2)
    ==>   z(1) = z(2)^2-z(2)^3 + ..... (terms of degree >=4)
    ==>
    ==> // Hamburger-Noether development of branch nr.3:
    ==>   y = z(1)*x^2
    ==>   x = z(1)^2
    ==>
    print(extdevelop(Hne[1],5)[1]);
    ==>  No extension is possible
    ==> 0,x,0,
    ==> 0,1,x
    list ehne=extdevelop(Hne[2],5);
    displayHNE(ehne);
    ==>   y = z(1)*x
    ==>   x = z(1)^2+z(1)^2*z(2)
    ==>   z(1) = z(2)^2-z(2)^3+z(2)^4-z(2)^5 + ..... (terms of degree >=6)
    param(Hne[2]);
    ==> // ** Warning: result is exact up to order 5 in x and 7 in y !
    ==> _[1]=x7-x6-x5+x4
    ==> _[2]=-x10+2x9-2x7+x6
    param(ehne);
    ==> // ** Warning: result is exact up to order 7 in x and 9 in y !
    ==> _[1]=x11-x10+x9-x8-x7+x6-x5+x4
    ==> _[2]=-x16+2x15-3x14+4x13-2x12+2x10-4x9+3x8-2x7+x6

See also: develop; hnexpansion; param.

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