By Shaked, Levy, Baharav, Yen
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For a polynomial with numerical coeﬃcients in the rational-number ﬁeld Q, the inﬁma can be algebraic numbers of degrees 1, 2 or 3. If the formulae (13) and (14) are used for substitution, the answer will always appear to be an algebraic Unconstrained Parametric Minimization of a Polynomial 29 number of degree 3, and the simpliﬁcation of such numbers into lower degree forms cannot be relied on in some systems. Therefore, if it is accepted that the system should return the simplest expressions possible, then the best strategy in this case is not to use (14), but instead to solve the cubic equation (17) directly.
Hence, it is important to decide whether there exists a real multiple zero since it may become close real zeros or complex zeros through perturbations of coeﬃcients, however small those perturbations may be. In particular, we consider the following type of problem. Given f ∈ R[x] and a closed real interval I, ﬁnd a nearest polynomial f˜ ∈ R[x] in a weighted l∞ -norm that has a multiple zero in I. In this paper, the lp -norm of a polynomial is the p-norm of the vector of coeﬃcients of the polynomial with respect to a given basis.
46 M. Safey El Din Theorem 1. Let f be a polynomial in Q[X1 , . . , Xn ] of degree D. The set of generalized critical values K(f ) of the mapping f : x ∈ Cn → f (x) ∈ C is Zariski-closed in C. Moreover, D K∞ (f ) + K0 (f ) ≤ Dn − 1 The main interest of the set of generalized critical values relies on its topological properties which are summarized below and proved in . Theorem 2. The mapping fC : x ∈ Cn → f (x) ∈ C realizes a locally trivial ﬁbration in Cn \ fC−1 (K(fC )). The mapping fR : x ∈ Rn → f (x) ∈ R realizes a locally trivial ﬁbration in Rn \ fR−1 (K(fR )).