Shub–Smale's polynomial expected-length conjecture for random start systems

Let $f$ be the target polynomial system, let $g$ be a start system chosen from the sphere ${\mathbb S}$ with the uniform probability distribution, and let $e_0$ be the initial solution specified by

$$g=\begin{cases}\sqrt{d_1}X_0^{d_1-1}X_1\\ \vdots\\ \sqrt{d_n}X_0^{d_n-1}X_n\end{cases},\qquad e_0=\begin{pmatrix}1\\0\\ \vdots\\0\end{pmatrix}.$$

Write $\mathcal{C}_0(f,g,e_0)$ for the length of the homotopy path in the condition metric, and let $N$ denote the dimension parameter used for the polynomial system. Shub–Smale's conjecture. For $g$ picked in the sphere ${\mathbb S}$ with the uniform probability distribution, the expected length of the homotopy path ${\rm E}(\mathcal{C}_0(f,g,e_0))$ is bounded by a polynomial in $N$. This conjecture concerns the average complexity of numerical homotopy continuation and would provide a polynomial expected-time bound for the corresponding random-start procedure; its resolution is not supplied in the source.