I am working with a family of polynomials with some particularly nasty coefficients. Unfortunately, I cannot give very many details as to what these polynomials look like, but, suffice it to say that I have some general polynomials
$$p_i(x) = a_{i,0}+\cdots+a_{i,n}x^{n_i}$$
and I would like to determine if there are any real zeros on the interval $[0,1]$. Explicitly computing the zeros of every member of this family of polynomials is infeasible, so what I am looking for is a list of techniques to determine a lower bound on the REAL zeros of these polynomials. Unfortunately, the naive approach using things like Lagrange's and Cauchy's bound for all of the zeros does not bear any fruit since there are complex zeros of some of these polynomials lie within the unit disk (though for the few thousand that I have explicitly computed zeros for, none of the real zeros are in the unit disk, and they, in fact, tend to drift away from the unit disk as the degree increases). Also, things like the Gershgorin circle theorem are a bit difficult to work with for this particular family (though, other spectral methods which I am heretofore unaware of might be more effective). So I am looking for other techniques that might be more fruitful but which are sufficiently general as to apply to a whole family of polynomials. Any help is greatly appreciated, thank you!