Rigid continuation paths I. Quasilinear average complexity for solving polynomial systems

Abstract

How many operations do we need on 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`op1q . This improves upon the previously known (input size)

Smale 17th Problem: Advances and Open Directions

“…where D k ζ F is the kth derivative of F at ζ and the norm is the operator norm. The γ-Theorem of Smale states that if F (ζ) = 0 and d P (z, ζ)γ(F, ζ) ≤ 1 6 then z is an approximate zero of F with associated zero ζ (see, e.g., [17,Thm. 12]).…”

“…An optimal answer was provided by Lairez [17]. The involved ideas were new and touched both the algorithm and its analysis.…”

“…for some constant C. Furthermore, Lairez [17,Prop. 17] shows that, for a given F ∈ H d with square-free components,…”

“…It shows that the average number of steps in the rigid homotopy is polynomial in n and D. It only remains to be seen that the cost of each such step is still linear in N . Lairez [17,Cor. 34] shows that this is the case: the cost of each step is O(n 2 D 2 N ).…”

Probabilistic Condition Number Estimates for Real Polynomial Systems I: A Broader Family of Distributions

“…To be more precise, the state of the art is represented by Lairez's recent article [30], which proves an expected complexity bound of O(n 6 d 4 N), under the preceding randomness model, where d := max i d i . The only drawback of these elegant results is that, in practice, the input size is often much smaller than N.…”