Singular

D.2.4.4 pdivi

Usage:

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

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.G), and no lpp of a divisor divides a pp of r.

LIB "grobcov.lib";
ring R=(0,a,b,c),(x,y),dp;
poly f=(ab-ac)*xy+(ab)*x+(5c);
// Divisor=";
f;
==> (ab-ac)*xy+(ab)*x+(5c)
ideal F=ax+b,cy+a;
// Dividends=";
F;
==> F[1]=(a)*x+(b)
==> F[2]=(c)*y+(a)
def r=pdivi(f,F);
// (Remainder, quotients, factor)=";
r;
==> [1]:
==>    (ab2-abc-b2c+5c2)
==> [2]:
==>    _[1]=(bc-c2)*y+(bc)
==>    _[2]=(-b2+bc)
==> [3]:
==>    (c)
// Verifying the division:
r[3]*f-(r[2][1]*F[1]+r[2][2]*F[2]+r[1]);
==> 0