## 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 $\text{(input}\phantom{\rule{4.pt}{0ex}}{\text{size)}}^{1+o\left(1\right)}$,
which improves upon the previously known $\text{(input}\phantom{\rule{4.pt}{0ex}}{\text{size)}}^{3/2+o\left(1\right)}$ 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.