Strong bi-homogeneous Bézout theorem and its use in effective real algebraic geometry

Let (f1, . . . , fs) be a polynomial family in Q[X1, . . . , Xn] (with s 6 n− 1) of degree bounded by D. Suppose that 〈f1, . . . , fs〉 generates a radical ideal, and defines a smooth algebraic variety V ⊂ Cn. Consider a projection π : Cn → C. We prove that the degree of the critical locus 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 Lagrange’s system for which a bihomogeneous structure is exhibited. Secondly we prove a bi-homogeneous Bézout Theorem, which bounds the sum of the degrees of the equidimensional components of the radical of an ideal generated by a bi-homogeneous polynomial family. This result is improved when (f1, . . . , fs) is a regular sequence. Moreover, we use Lagrange’s system to design an algorithm computing at least one point in each connected component of a smooth real algebraic set. This algorithm generalizes, to the non equidimensional case, the one of Safey El Din and Schost. The evaluation of the output size of this algorithm gives new upper bounds on the first Betti number of a smooth real algebraic set. Finally, we estimate its arithmetic complexity and prove that in the worst cases it is polynomial in n, s, Ds(D − 1)n−s ( n n−s ) and the complexity of evaluation of f1, . . . , fs.