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D.12.6.1 weierstrDiv
Procedure from library weierstr.lib (see weierstr_lib).
- Usage:
- weierstrDiv(g,f,d); g,f=poly, d=integer
- Assume:
- f must be general of finite order, say b, in the last ring variable,
say T; if not use the procedure lastvarGeneral first
- Purpose:
- perform the Weierstrass division of g by f up to order d
- Return:
- - a list, say l, of two polynomials and an integer, such that
g = l[1]*f + l[2], deg_T(l[2]) < b, up to (including) total degree d
- l[3] is the number of iterations used
- if f is not T-general, return (0,g)
- 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 = y - xy2 + x2;
poly g = y;
list l = weierstrDiv(g,f,10); l;"";
==> [1]:
==> 1+xy-x3+x2y2-2x4y+2x6+x3y3-3x5y2+5x7y+x4y4-5x9-4x6y3+9x8y2+x5y5
==> [2]:
==> -x2+x5-2x8
==> [3]:
==> 5
==>
l[1]*f + l[2]; //g = l[1]*f+l[2] up to degree 10
==> y-5x11+14x10y2+5x7y5-9x9y4-x6y7
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