How many operations do we need on the average to compute an approximate root
of a random Gaussian polynomial system? Beyond Smale's 17th problem that asked
whether a polynomial bound is possible, we prove a quasi-optimal bound
(input size)1+o(1). This improves upon the previously known
(input size)23+o(1) bound.
The new algorithm relies on numerical continuation along \emph{rigid
continuation paths}. The central idea is to consider rigid motions of the
equations rather than line segments in the linear space of all polynomial
systems. This leads to a better average condition number and allows for bigger
steps. We show that on the average, we can compute one approximate root of a
random Gaussian polynomial system of~n equations of degree at most D in
n+1 homogeneous variables with O(n5D2) continuation steps. This is a
decisive improvement over previous bounds that prove no better than
2min(n,D) continuation steps on the average