Singular

D.2.4.4 pdivi

Usage:

pdivi(poly f,ideal F);
poly f: the polynomial in Q[a][x] to be divided
ideal F: the divisor ideal in Q[a][x].
(a=parameters, x=variables).

Return:

A list (poly r, ideal q, poly m). r is the remainder of the pseudodivision, q is the set of quotients, and m is the coefficient factor by which f is to be multiplied.

Note:

pseudodivision of a poly f by an ideal F in Q[a][x]. Returns a list (r,q,m) such that m*f=r+sum(q.F),
and no lpp of a divisor divides a pp of r.

LIB "grobcov.lib";
if(defined(R)){kill R;}
ring R=(0,a,b,c),(x,y),dp;
short=0;
// Divisor=";
poly f=(ab-ac)*xy+(ab)*x+(5c);
// Dividends=";
ideal F=ax+b,cy+a;
// (Remainder, quotients, factor)=";
def r=pdivi(f,F); r;
==> [1]:
==>    (a*b^2-a*b*c-b^2*c+5*c^2)
==> [2]:
==>    _[1]=(b*c-c^2)*y+(b*c)
==>    _[2]=(-b^2+b*c)
==> [3]:
==>    (c)
// Verifying the division:
r[3]*f-(r[2][1]*F[1]+r[2][2]*F[2]+r[1]);
==> 0