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Your data is fishy:

sage: denom.parent().term_order()
Negative weighted degree reverse lexicographic term order with weights (1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4)

This 'local' term ordering causes reduce and (hence) divides to behave differently. In particular, it does not give the same result as with a well-ordering such as 'lex' in your ring dR:

sage: dR(denom).divides(dR(num))
False

So the polynomials do not divide each other in the ordinary sense. Maybe they do in some 'local' sense, but I don't know anything about local orderings other than their existence. I would be glad if someone could explain it.

Your data is fishy:

sage: denom.parent().term_order()
Negative weighted degree reverse lexicographic term order with weights (1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4)

This 'local' term ordering causes reduce and (hence) divides to behave differently. In particular, it does not give the same result as with a well-ordering such as 'lex' in your ring dR:

sage: dR(denom).divides(dR(num))
False

So the polynomials do denom does not divide each other num in the ordinary sense. Maybe they do in some 'local' sense, but I don't know anything about local orderings other than their existence. I would be glad if someone could explain it.sense.