Bill Hart wrote on [sage-devel]:
"""
Almost everything over Q should probably be converted to a problem
over Z. I haven't seen any polynomial problems over Q which should not
be dealt with this way so far, but I suppose they may exist.
"""
Further justification:
sage: f = R.random_element(2000)
sage: g = R.random_element(2000)
sage: fD = f.denominator()
sage: gD = g.denominator()
sage: fZ = (fD * f).change_ring(ZZ)
sage: gZ = (gD * g).change_ring(ZZ)
sage: %time _ = f*g
CPU times: user 0.63 s, sys: 0.02 s, total: 0.66 s
Wall time: 0.67 s
sage: %time _ = (fZ*gZ)
CPU times: user 0.01 s, sys: 0.00 s, total: 0.01 s
Wall time: 0.01 s
sage: %time _ = (fZ*gZ)/(fD*gD)
CPU times: user 0.06 s, sys: 0.00 s, total: 0.06 s
Wall time: 0.06 s
sage: fM = magma(f)
sage: gM = magma(g)
sage: t = magma.cputime()
sage: _ = fM*gM
sage: magma.cputime(t)
0.059999999999999998
sage: f = R.random_element(4000)
sage: g = R.random_element(4000)
sage: fD = f.denominator()
sage: gD = g.denominator()
sage: fZ = (fD * f).change_ring(ZZ)
sage: gZ = (gD * g).change_ring(ZZ)
sage: %time _ = f*g
CPU times: user 2.11 s, sys: 0.00 s, total: 2.12 s
Wall time: 2.14 s
sage: %time _ = (fZ*gZ)
CPU times: user 0.02 s, sys: 0.00 s, total: 0.02 s
Wall time: 0.02 s
sage: %time _ = (fZ*gZ)/(fD*gD)
CPU times: user 0.14 s, sys: 0.01 s, total: 0.15 s
Wall time: 0.15 s
sage: fM = magma(f)
sage: gM = magma(g)
sage: t = magma.cputime()
sage: _ = fM*gM
sage: magma.cputime(t)
0.10000000000000001
