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D.6.11.6 invariants
Procedure from library hnoether.lib (see hnoether_lib).
- Usage:
- invariants(INPUT); INPUT list or poly
- Assume:
INPUT is the output of develop(f), or of
extdevelop(develop(f),n), or one entry of the list of HN data
computed by hnexpansion(f[,"ess"]).
- Return:
- list
INV of the following format:
| | INV[1]: intvec (characteristic exponents)
INV[2]: intvec (generators of the semigroup)
INV[3]: intvec (Puiseux pairs, 1st components)
INV[4]: intvec (Puiseux pairs, 2nd components)
INV[5]: int (degree of the conductor)
INV[6]: intvec (sequence of multiplicities)
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If INPUT contains no valid HN expansion, the empty list is
returned.
- Assume:
INPUT is a bivariate polynomial f, or the output of
hnexpansion(f), or the list of HN data computed by
hnexpansion(f [,"ess"]).
- Return:
- list
INV, such that INV[i] coincides with the output of
invariants(develop(f[i])), where f[i] is the i-th branch of
f, and the last entry of INV contains further invariants of
f in the format:
| | INV[last][1] : intmat (contact matrix of the branches)
INV[last][2] : intmat (intersection multiplicities of the branches)
INV[last][3] : int (delta invariant of f)
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- Note:
- In case the Hamburger-Noether expansion of the curve f is needed
for other purposes as well it is better to calculate this first
with the aid of
hnexpansion and use it as input instead of
the polynomial itself.
Example:
| | LIB "hnoether.lib";
ring exring=0,(x,y),dp;
list Hne=develop(y4+2x3y2+x6+x5y);
list INV=invariants(Hne);
INV[1]; // the characteristic exponents
==> 4,6,7
INV[2]; // the generators of the semigroup of values
==> 4,6,13
INV[3],INV[4]; // the Puiseux pairs in packed form
==> 3,7 2,2
INV[5] div 2; // the delta-invariant
==> 8
INV[6]; // the sequence of multiplicities
==> 4,2,2,1,1
// To display the invariants more 'nicely':
displayInvariants(Hne);
==> characteristic exponents : 4,6,7
==> generators of semigroup : 4,6,13
==> Puiseux pairs : (3,2)(7,2)
==> degree of the conductor : 16
==> delta invariant : 8
==> sequence of multiplicities: 4,2,2,1,1
/////////////////////////////
INV=invariants((x2-y3)*(x3-y5));
INV[1][1]; // the characteristic exponents of the first branch
==> 2,3
INV[2][6]; // the sequence of multiplicities of the second branch
==> 3,2,1,1
print(INV[size(INV)][1]); // the contact matrix of the branches
==> 0 3
==> 3 0
print(INV[size(INV)][2]); // the intersection numbers of the branches
==> 0 9
==> 9 0
INV[size(INV)][3]; // the delta invariant of the curve
==> 14
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See also:
develop;
displayInvariants;
hnexpansion;
intersection;
multsequence.
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