For a proof, I need to check that given a little interval $(0, 0.28)$ some concrete polynomials $\in \mathbb{Q}[w]$ (polynomials in one variable ranging over the real numbers, with degrees around 50) are positive there, or equivalently that they have no roots in the same interval (since evaluating in 0 I get a positive value).
Taking this into account, what would you recommend me to do? I was considering Sturm algorithm, but I am open to suggestions and orientation.
As for the language, I had thought of Python, but I wouldn't mind using Sage, C, Java...