  Smale's 17th problem asks for an algorithm which finds an approximate zero of
polynomial systems in average polynomial time (see Smale 2000). The main
progress on Smale's problem is Beltr\'an-Pardo (2011) and B\"urgisser-Cucker
(2010). In this paper we will improve on both approaches and we prove an
important intermediate result. Our main results are Theorem 1 on the complexity
of a randomized algorithm which improves the result of Beltr\'an-Pardo (2011),
Theorem 2 on the average of the condition number of polynomial systems which
improves the estimate found in B\"urgisser-Cucker (2010), and Theorem 3 on the
complexity of finding a single zero of polynomial systems. This last Theorem is
the main result of B\"urgisser-Cucker (2010). We give a proof of it relying
only on homotopy methods, thus removing the need for the elimination theory
methods used in B\"urgisser-Cucker (2010). We build on methods developed in
Armentano et al. (2015).
