It is rather frustrating that sage does not a have a gcd algorithm
for
multivariate polynomials over generic fields.

There is such an algorithm. It is called the "subresultant" algorithm.
It is for example described
in Knuth's book "Semi-numerical algorithms" in section 4.6.1.

Does anybody have experience with this algorithm? Does it perform
acceptably? I think Magma
uses it as a fallback if more specialized algorithms fail.

Michel


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