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

Strong bi-homogeneous Bézout theorem and its use in effective real algebraic geometry,Computing Research Repository,Mohab Safey El Din,Philippe Trebuc

Strong bi-homogeneous Bézout theorem and its use in effective real algebraic geometry   (Citations: 3)
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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 Cn. Consider a projection : Cn ! C. We prove that the degree of the critical locus of 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 restricted to V are characterized as projections of the solutions of Lagrange's system for which a bi- homogeneous structure is exhibited. Secondly we prove a bi-homogeneous B ezout Theorem, which bounds the sum of the degrees of the equidimensional compo- nents 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 con- nected 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, Ds(D 1)n s n n s and the complexity of evaluation of f1; : : : ; fs.
Journal: Computing Research Repository - CORR , vol. abs/cs/061, 2006
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    • ...We make use of the results of [19] which shows how to use Lagrange’s system in conjunction with Lecerf’s results [16] to improve the complexity of computing critical points and [16] which bounds the complexity of computing a lifted curve as a parametrized geometric resolution of 1-dimensional variety dened by polynomials of degree D by D O(n) ...

    Marc MEZZAROBBA. Computing Roadmaps in Smooth Real Algebraic Sets

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