In this paper, we propose a silent self-stabilizing leader election algorithm for bidirectional connected identified networks of arbitrary topology. This algorithm is written in the locally shared memory model. It assumes the distributed unfair daemon, the most general scheduling hypothesis of the model. Our algorithm requires no global knowledge on the network (such as an upper bound on the diameter or the number of processes, for example). We show that its stabilization time is in Θ(n3) steps in the worst case, where n is the number of processes. Its memory requirement is asymptotically optimal, {\em i.e.}, Θ(logn) bits per processes. Its round complexity is of the same order of magnitude --- {\em i.e.}, Θ(n) rounds --- as the best existing algorithm (Datta et al, 2011) designed with similar settings ({\em i.e.}, it does not use global knowledge and is proven under the unfair daemon). To the best of our knowledge, this is the first self-stabilizing leader election algorithm for arbitrary identified networks that is proved to achieve a stabilization time polynomial in steps. By contrast, we show that the previous best existing algorithm designed with similar settings (Datta et al, 2011) may stabilize in a non polynomial number of steps.