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D.5.5.9 multsequence
Procedure from library hnoether.lib (see hnoether_lib).
- Usage:
- multsequence(L); L list
- Assume:
- L is the output of
develop(f) , or of
extdevelop(develop(f),n) , or one entry of the output of
reddevelop(f) .
- Return:
- intvec corresponding to the multiplicity sequence of (a branch)
of the curve (the same as
invariants(L)[6] ).
- Assume:
- L is the output of
reddevelop(f) .
- Return:
- list of two integer matrices:
multsequence(L)[1][i,*]
- contains the multiplicities of the branches at their infinitely near point
of 0 in its (i-1) order neighbourhood (i.e., i=1: multiplicity of the
branches themselves, i=2: multiplicity of their 1st quadratic transformed,
etc.,
Hence, multsequence(L)[1][*,j] is the multiplicity sequence
of branch j.
multsequence(L)[2][i,*] :
- contains the information which of these infinitely near points coincide.
- Note:
- The order of elements of the list obtained from
reddevelop(f)
must not be changed (because then the coincident infinitely near points
couldn't be grouped together, cf. meaning of 2nd intmat in example).
Hence, it is not wise to compute the HNE of several polynomials
separately, put them into a list L and call multsequence(L) .
Use displayMultsequence to produce a better readable output for
reducible curves on the screen.
Example:
| LIB "hnoether.lib";
// -------- prepare for example ---------
if (nameof(basering)=="HNEring") {
def rettering=HNEring;
kill HNEring;
}
// ------ the example starts here -------
int plevel=printlevel; printlevel=-1;
ring r=0,(x,y),dp;
list hne=reddevelop((x6-y10)*(x+y2-y3)*(x+y2+y3)); // 4 branches
multsequence(hne[1])," | ",multsequence(hne[2])," | ",
multsequence(hne[3])," | ",multsequence(hne[4]);
==> 3,2,1 | 3,2,1 | 1 | 1
multsequence(hne);
==> [1]:
==> 3,3,1,1,
==> 2,2,1,1,
==> 1,1,1,1,
==> 1,1,1,1,
==> 1,1,1,1
==> [2]:
==> 4,0,0,0,
==> 4,0,0,0,
==> 2,2,0,0,
==> 2,1,1,0,
==> 1,1,1,1
// The meaning of the entries of the 2nd matrix is as follows:
displayMultsequence(hne);
==> [(3,3,1,1)],
==> [(2,2,1,1)],
==> [(1,1),(1,1)],
==> [(1,1),(1),(1)],
==> [(1),(1),(1),(1)]
kill HNEring,r;
printlevel=plevel;
// --- restore HNEring if previously defined ---
if (defined(rettering)) {
setring rettering;
def HNEring=rettering;
export HNEring;
}
| See also:
develop;
displayMultsequence;
reddevelop;
separateHNE.
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