D.12.5.1 weierstrDiv

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

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