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Cheta with base-change: preliminary results
I also just see that there is a function lagrange_polynomial in sage
# using the definition of Lagrange interpolation polynomial
sage: R = PolynomialRing(QQ, 'x')
sage: R.lagrange_polynomial([(0,1),(2,2),(3,-2),(-4,9)])

I mean this should be super easy now. Just plug in your argument-value-pairs and you have the interpolating polynomial (no matrix fuzz).
Then you can apply this interpolating polynomial to non-real values.
Or extract the coefficients as you like.
However I didnt check how long it takes Smile
For variants see

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