Hi!

Here is a first ticket, with patch: 
http://trac.sagemath.org/sage_trac/ticket/7578

It reduces the computation time of the original example of this thread
to the time that is needed to convert the underlying finite
polynomials.  Hence, as long as the conversion does not improve, it
seems to me that the arithmetic of the infinite polynomials can not be
further improved.

Or, positively: There seems to be considerable room for improving the
polynomial conversion, and the infinite polynomials would directly
profit from any progress. But this should be a different ticket.

Cheers,
Simon

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