1. Strong Bi-homogeneous Bézout's Theorem and degree bounds for algebraic optimization
- Author
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Safey El Din, Mohab, Trebuchet, Philippe, Solving problems through algebraic computation and efficient software (SPACES), INRIA Lorraine, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Lorrain de Recherche en Informatique et ses Applications (LORIA), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Systèmes Polynomiaux, Implantation, Résolution Algébrique (SPIRAL), Laboratoire d'Informatique de Paris 6 (LIP6), Université Pierre et Marie Curie - Paris 6 (UPMC)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Centre National de la Recherche Scientifique (CNRS), INRIA, Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP), and Université Pierre et Marie Curie - Paris 6 (UPMC)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
REAL SOLUTIONS ,Mathematics::Commutative Algebra ,POLYNOMIAL SYSTEMS ,[INFO.INFO-OH]Computer Science [cs]/Other [cs.OH] ,Computer Science::Symbolic Computation - Abstract
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.
- Published
- 2004