Abstract: In a series of papers written in the first half of the
1990's Steve Smale and I studied the complexity of solving systems of
n polynomial equations in n complex variables. We studied path
following techniques. A system with known solution is connected by a
path to the system we want to solve and the solution is "continued"
along the path. The path we chose was the straight line connecting the
systems. We proved that "on average" systems can be solved with
polynomial cost but we did not prove the existence of a uniform
algorithm. The question of the existence of a uniform algorithm is
Smale's 17th problem. Recently, Beltran and Pardo have made
significant progress on this problem. Moreover, Jointly with Beltran I
have linked the complexity to the length of the (problem,solution)
path in the condition number Riemannian structure. Surprisingly short
paths exist! So the study of the geodesics of this Riemannian
structure presents interesting challenges.