Summary: We present a solution to the point location problem in arrangements of hyperplanes in \(E^ d\) with running time \(O(d^ 5\log n)\) and space \(O(n^{d+\kappa})\) for arbitrary \(\kappa>0\), where \(n\) is the number of hyperplanes. The main result is the \(d^ 5\) factor in the expression for the running time. All previously known algorithms are exponential in \(d\) or \(\log n\). This leads to nonuniform polynomial algorithms for NP- complete problems.