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D.5.5.5 parametrisation
Procedure from library hnoether.lib (see section hnoether_lib).
- Usage:
parametrisation(INPUT [,x]); INPUT list or poly, x int (optional)
- Assume:
INPUT is either a bivariate polynomial f defining a plane curve
singularity, or it is the output of hnexpansion(f[,"ess"]) ,
or of develop(f) , or of extdevelop(develop(f),n) ,
or the list @{hne} in the ring created by hnexpansion(f)
respectively one entry thereof.
- Return:
a list L containing a parametrization L[i] for each branch f[i] of f
in the following format:
- if only the list INPUT is given, L[i] is an ideal of two polynomials
p[1],p[2]: if the HNE of was finite then f[i](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, L[i] is itself a list:
L[i][1] is the parametrization ideal as above and L[i][2] is an intvec
with two entries indicating the highest degree up to which the
coefficients of the monomials in L[i][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.
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,t),ds;
// 1st Example: input is a polynomial
poly g=(x2-y3)*(x3-y5);
parametrisation(g);
→ [1]:
→ _[1]=t3
→ _[2]=t2
→ [2]:
→ _[1]=t5
→ _[2]=t3
// 2nd Example: input is the ring of a Hamburger-Noether expansion
poly h=x2-y2-y3;
list hn=hnexpansion(h);
parametrisation(h,1);
→ [1]:
→ [1]:
→ _[1]=t
→ _[2]=t-1/2t2
→ [2]:
→ -1,2
→ [2]:
→ [1]:
→ _[1]=t
→ _[2]=-t-1/2t2
→ [2]:
→ -1,2
// 3rd Example: input is a Hamburger-Noether expansion
poly f=x3+2xy2+y2;
list hne=develop(f);
list hne_extended=extdevelop(hne,10);
// 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:
parametrisation(hne);
→ [1]:
→ _[1]=-t2
→ _[2]=-t3
parametrisation(hne_extended);
→ [1]:
→ _[1]=-t2+2t4-4t6+8t8-16t10
→ _[2]=-t3+2t5-4t7+8t9-16t11
parametrisation(hne_extended,0);
→ [1]:
→ [1]:
→ _[1]=-t2+2t4-4t6+8t8-16t10
→ _[2]=-t3+2t5-4t7+8t9-16t11
→ [2]:
→ 10,11
See also:
develop;
extdevelop.
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