Nonlinear Equations - UFRJ

Chapter 10
Homotopy
Several recent breakthroughs made Smale’s 17 th problem
an active, fast-moving subject. The first part of the Bézout saga
[70–74] culminated in the existential proof of a non-uniform, average
polynomial time algorithm to solve Problem 1.11. Namely,
Theorem 10.1 (Shub and Smale). Let H d be endowed with the normal
(Gaussian) probability distribution dH d with mean zero and variance
1.
There is a constant c such that, for every n, for every d =
(d 1 , . . . , d n ), there is an algorithm to find an approximated root of a
random f ∈ (H d , dH d ) within expected time cN 4 , where N = dim H d
is the input size.
This theorem was published in 1994, and motivated the statement
of Smale’s 17 th problem. It was obtained through the painful complexity
analysis of a linear homotopy method. Given F 0 , F 1 ∈ H d
and x 0 and approximate zero of F 0 , the homotopy method was of the
Gregorio Malajovich, Nonlinear equations.
28 o Colóquio Brasileiro de Matemática, IMPA, Rio de Janeiro, 2011.
Copyright c○ Gregorio Malajovich, 2011.
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