Shub–Smale's optimal-start-pair conjecture

Let $V=\{(h,z)\in{\mathbb S}\times{\mathbb P}({\mathbb C}^{n+1})\mid h(z)=0\}$ be the solution variety, and let $(g,e_0)$ be the initial pair with

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

For an initial pair $(g,z_0)\in V$, write $\mathcal{C}_0(f,g,z_0)$ for the homotopy-path length. Shub–Smale's optimal-start-pair conjecture. The initial pair $(g,e_0)$ as described above minimizes ${\rm E}(\mathcal{C}_0(f,g,z_0))$ over all initial pairs $(g,z_0)$ in the solution variety $V$. This stronger conjecture is motivated by the paper's computational experiments, which found no initial pair outperforming $(g,e_0)$ in the tested cases; the general minimization claim remains open in the supplied source.