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D.12.5.2 weierstrPrep
Procedure from library weierstr.lib (see weierstr_lib).
- Usage:
- weierstrPrep(f,d); f=poly, d=integer
- Assume:
- f must be general of finite order, say b, in the last ring variable,
say T; if not apply the procedure lastvarGeneral first
- Purpose:
- perform the Weierstrass preparation of f up to order d
- Return:
- - a list, say l, of two polynomials and one integer,
l[1] a unit, l[2] a Weierstrass polynomial, l[3] an integer
such that l[1]*f = l[2], where l[2] is a Weierstrass polynomial,
(i.e. l[2] = T^b + lower terms in T) up to (including) total degree d
l[3] is the number of iterations used
- if f is not T-general, return (0,0)
- Note:
- the procedure works for any monomial ordering
- Theory:
- the proof of Grauert-Remmert (Analytische Stellenalgebren) is used
for the algorithm
Example:
| LIB "weierstr.lib";
ring R = 0,(x,y),ds;
poly f = xy+y2+y4;
list l = weierstrPrep(f,5); l; "";
==> [1]:
==> 1-x2+xy-y2+3x4-3x3y+3x2y2-2xy3+y4
==> [2]:
==> xy+y2-x3y
==> [3]:
==> 6
==>
f*l[1]-l[2]; // = 0 up to degree 5
==> 3x5y+3x4y4-3x3y5+3x2y6-2xy7+y8
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