Before revision:  sets from \mathcal{S . Such a polynomial is called a " generalized partitioning polynomial " . We present a randomized algorithm that efficiently computes such a polynomial P. Specifically, the expected running time of our algorithm is only linear in | \mathcal{S | , where the constant of proportionality depends on d, D, and the complexity of the description of~\mathcal{S . Our approach exploits the technique of "quantifier elimination " combined with that of " \epsilon-samples ". We present four applications of our result.
After revision:  sets from \mathcal{S . Such a polynomial is called a " generalized partitioning polynomial " . We present a randomized algorithm that efficiently computes such a polynomial P. Specifically, the expected running time of our algorithm is only linear in | \mathcal{S | , where the constant of proportionality depends on d, D, and the complexity of the description of~\mathcal{S . Our approach exploits the technique of "quantifier elimination " combined with that of " \epsilon-samples . We also present an extension of our construction to multi-level polynomial partitioning for semi-algebraic sets in \mathbb{R applications of our result.