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• W2788179600 abstract "Let (f1; : : : ; fs) be a polynomial family in Q[X1; : : : ; Xn] (with s 6 n 1) of degree bounded by D. Suppose that hf1; : : : ; fsi generates a radical ideal, and denes a smooth algebraic variety V C n . Consider a projection : C n ! C. We prove that the degree of the critical locus of restricted to V is bounded by D s (D 1) n s n n s . This result is obtained in two steps. First the critical points of 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 B ezout 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 rst 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, D s (D 1) n s n n s and the complexity of evaluation of f1; : : : ; fs." @default.
• W2788179600 created "2018-03-06" @default.
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• W2788179600 date "2006-01-01" @default.
• W2788179600 modified "2023-09-26" @default.
• W2788179600 title "Strong bi-homogeneous B ezout theorem and its use in eectiv e real algebraic geometry" @default.
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