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I see that I misunderstood the words "generic" and "with_category" in type name "sage.rings.polynomial.polynomial_element_generic.PolynomialRing_field_with_category.element_class". I thought the type name means something like a common superclass to the various implementations ("generic") with a switch to the actual implementation ("with_category"). Now I see that the type name specifies a third implementation besides FLINT and NTL. I read documentation "Sage Reference Manual: Polynomials" forth and back but I did not find a hint on this meaning. Would it be possible to add a relating comment into this paper (and into the "Tutorial")? Moreover, it may be wise to add a comment into the online documentation of "PolynomialRing.__init__" stating that argument "implementation" is ignored if "sparse=True" and a third (not otherwise mentioned) implementation is used instead.

Thanks for the hint to use multivariate polynomials instead of the univariate ones. Polynomials of a degree so big that they do not fit into memory when using a dense univariate implementation do not make problems anymore. But, of course, the multivariate implementation adds some computational overhead. For my polynomials when the dense univariate implementation is on its limits (degree about 10^7, density 1.2%), the computing times for polynomial generation (most time is spent for multiplication) are coarsely related as

univariate_dense : multivariate : univariate_sparse = 1 : 500 : 1000.

So switching to multivariate polynomials does not much help. In other words, the implementation of sparse univariate polynomials by a Python dictionary is not as bad as one might think.