I have an integer coefficient polynomial, $f$. I want to check if for any $x > 1$, $f(x) < 0$. Some ideas involve Descartes' rule of sign and Sturm's theorem. However, I would like to do this on a GPU! Therefore, both these methods are not ideal. Sturm's theorem requires performing a polynomial modulo. Both ideas involve iteration over the roots.
For those unfamiliar, GPUs would like many calculations that can be done at the same time with the same operation. For example, evaluating a polynomial at many different input points would perform the exact same calculations many times, just on different input. Other fast operations include matrix multiplication, vector addition, summing tensors across a subset of dimensions, and so on.
I am wondering if there are some alternate approaches out there - ones that may not involve finding the roots. Even if you don't know if the approach can be put on GPU, ideas are very much appreciated.