Section: New Results

Quasilinear Average Complexity for Solving Polynomial Systems

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, Pierre Lairez has proved in [10] a quasi-optimal bound $\text{(input}{\text{size)}}^{1+o\left(1\right)}$, which improves upon the previously known $\text{(input}{\text{size)}}^{3/2+o\left(1\right)}$ bound. His new algorithm relies on numerical continuation along 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. He showed that on the average, one approximate root of a random Gaussian polynomial system of $n$ equations of degree at most $D$ in $n+1$ homogeneous variables can be computed with $O\left(n^5{D}^2\right)$ continuation steps. This is a decisive improvement over previous bounds, which prove no better than ${\sqrt{2}}^{min(n,D)}$ continuation steps on the average.