Strong Bi-homogeneous Bézout’s Theorem and Degree Bounds for Algebraic Optimization

Let (f1, . . . , fs) be a polynomial family in Q[X1, . . . , Xn] (with s ≤ n − 1) of degree bounded by D, generating a radical ideal, and defining a smooth algebraic variety V ⊂ C. Consider a generic projection π : C → C, its restriction to V and its critical locus which is supposed to be zero-dimensional. We state that the number of critical points of π restricted to V is bounded by Ds(D−1)n−s ( n n−s ) . This result is obtained in two steps. First the critical points of π restricted to V are characterized as projections of the solutions of the Lagrange system for which a bi-homogeneous structure is exhibited. Next, we apply a strong 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 (f1, . . . , fs) 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, the evaluation of the size of the output of this algorithm gives new upper bounds on the first Betti number of a smooth real algebraic set.