We investigate a standard operator on classes of languages: unambiguous polynomial closure. We show that if C is a class of regular languages having some mild properties, the membership problem for its unambiguous polynomial closure UPol(C) reduces to the same problem for C. We give a new, self-contained and elementary proof of this result. We also show that unambiguous polynomial closure coincides with alternating left and right deterministic closure. Finally, if additionally C is finite, we show that the separation and covering problems are decidable for UPol(C).