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We tried a bit more and it turns out that dividing by a small irreducible polynomial will actually work:

sturm(((2*k+1) *a + 1)/(k^2+1))
=> [((2*k + 1)/(k^2 + 1))*a + 1/(k^2 + 1), (2*k + 1)/(k^2 + 1)]

The spurious factor cannot change the number of sign changes, so that's a workaround.