Strong Bi-homogeneous Bézout's Theorem and degree bounds for algebraic optimization

Let $(f_1, \ldots, f_s)$ be a polynomial family in $\Q[X_1, \ldots, X_n]$ (with $s\leq n-1$) of degree bounded by $D$, generating a radical ideal, and defining a smooth algebraic variety $\mathcal{V}\subset\C Consider a {\em generic} projection $\pi:\Cightarrow\Cts restriction to $\mathcal{V}$ and its critical locus which is supposed to be zero-dimensional. We state that the number of critical points of $\pi$ restricted to $\mathcal{V}$ is bounded by $D^s(D-1)^{n-s}{{n}\choose{n-s}}$. This result is obtained in two steps. First the critical points of $\pi$ restricted to $\mathcal{V}$ are characterized as projections of the solutions of the Lagrange system for which a bi-homogeneous structure is exhibited. Secondly we apply a bi-homogeneous Bézout Theorem, for which we give a proof and which bounds the sum of the degrees of the isolated primary components of an ideal generated by a bi-homogeneous family for which we give a proof. This result is improved in the case where $(f_1, \ldots, f_s)$ is a regular sequence. Moreover, we use Lagrange's system to generalize the algorithm due to Safey El Din and Schost for computing at least one point in each connected component of a smooth real algebraic set to the non equidimensional case. Then, evaluating the size of the output of this algorithm gives new upper bounds on the first Betti number of a smooth real algebraic set.