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

TL;DR

This paper introduces a new algorithm using rigid continuation paths to solve random Gaussian polynomial systems more efficiently, achieving a near-optimal average complexity bound of (input size)^{1+o(1)}.

Contribution

It presents a novel approach with rigid motions of equations, improving average condition number and step size, leading to a significant reduction in the number of steps needed.

Findings

Achieves an average complexity bound of (input size)^{1+o(1)}.

Uses rigid continuation paths to improve condition numbers and step sizes.

Reduces the number of steps to O(n^5 D^2) for solving polynomial systems.

Abstract

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 $\text{(input size)}^{1+o(1)}$. This improves upon the previously known $\text{(input size)}^{\frac32 +o(1)}$ 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 $O(n^5 D^2)$ continuation steps. This is a decisive improvement over previous bounds that prove no better than $\sqrt{2}^{\min(n, D)}$ continuation steps on the average.