Division algorithm in a polynomial ring with variable coefficients

edit

The following worked for me:

def div( f, g, ring ):
    """
    Division algorithm on Page 11 of Using AG by Cox;
    f is the dividend;
    g is a list of ordered divisors;
    The output consists of a list of coefficients for g and the remainder;
    p is the intermediate dividend;
    """
    n = len(g)
    p, r, q = f, 0,  [0 for _ in range(0,n)]

    count = 0
    while p != 0:
        count += 1
        i, divisionoccured = 0, False
        print ( "Step %s:\n\tp = %s\n\tr = %s\n\tq = %s"
                % ( count, p, r, q ) )

        while i < n and not divisionoccured:
            glt, plt = g[i].lt(), p.lt()

            if glt and plt / glt in ring:
                q[i] = q[i] + plt // glt
                p    =    p - plt // glt * g[i]
                divisionoccured = True
                print "\t\tChange:\n\t\tq[%s] = %s\n\t\tp = %s" % ( i, q[i], p )
            else:
                i = i + 1
                print "\t\tIncrementing i: i = %s" % i
        if not divisionoccured:
            r = r + p.lt()
            p = p - p.lt()

    return q, r

R.<a,b>    = PolynomialRing( QQ )
K          = FractionField( R )
RK.<x,y,z> = PolynomialRing( K, order='lex' )

f  = a*x^2*y^3 + x*y + 2*b
g1 = a^2*x + 2
g2 = x*y   - b

div( f, [g1,g2], RK )

This replaces the line with the divides method call by a test if the division (makes sens and) lands still in the ring.

Results with many verbose prints in the one test case submitted:

Step 1:
    p = a*x^2*y^3 + x*y + 2*b
    r = 0
    q = [0, 0]
        Change:
        q[0] = 1/a*x*y^3
        p = ((-2)/a)*x*y^3 + x*y + 2*b
Step 2:
    p = ((-2)/a)*x*y^3 + x*y + 2*b
    r = 0
    q = [1/a*x*y^3, 0]
        Change:
        q[0] = 1/a*x*y^3 + ((-2)/a^3)*y^3
        p = x*y + 4/a^3*y^3 + 2*b
Step 3:
    p = x*y + 4/a^3*y^3 + 2*b
    r = 0
    q = [1/a*x*y^3 + ((-2)/a^3)*y^3, 0]
        Change:
        q[0] = 1/a*x*y^3 + ((-2)/a^3)*y^3 + 1/a^2*y
        p = 4/a^3*y^3 + ((-2)/a^2)*y + 2*b
Step 4:
    p = 4/a^3*y^3 + ((-2)/a^2)*y + 2*b
    r = 0
    q = [1/a*x*y^3 + ((-2)/a^3)*y^3 + 1/a^2*y, 0]
        Incrementing i: i = 1
        Incrementing i: i = 2
Step 5:
    p = ((-2)/a^2)*y + 2*b
    r = 4/a^3*y^3
    q = [1/a*x*y^3 + ((-2)/a^3)*y^3 + 1/a^2*y, 0]
        Incrementing i: i = 1
        Incrementing i: i = 2
Step 6:
    p = 2*b
    r = 4/a^3*y^3 + ((-2)/a^2)*y
    q = [1/a*x*y^3 + ((-2)/a^3)*y^3 + 1/a^2*y, 0]
        Incrementing i: i = 1
        Incrementing i: i = 2
([1/a*x*y^3 + ((-2)/a^3)*y^3 + 1/a^2*y, 0], 4/a^3*y^3 + ((-2)/a^2)*y + 2*b)