M. Shub et S. Smale, COMPLEXITY OF BEZOUT THEOREM .4. PROBABILITY OF SUCCESS - EXTENSIONS, SIAM journal on numerical analysis, 33(1), 1996, pp. 128-148
We estimate the probability that a given number of projective Newton s
teps applied to a linear homotopy of a system of n homogeneous polynom
ial equations in n + 1 complex variables of fixed degrees will find al
l the roots of the system. We also extend the framework of our analysi
s to cover the classical implicit function theorem and revisit the con
dition number in this context. Further complexity theory is developed.