@MISC{Raffalli_distanceto, author = {C. Raffalli}, title = {DISTANCE TO THE DISCRIMINANT}, year = {} }

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Abstract

Abstract. The main contribution of this article is to establish that for an homogeneous polynomial P of degree d with n variables, every component of the complement of the 0 level of P in P n (R) contains a sphere whose radius is proportional to the square root of the distance between P and the discriminant (the set of polynomial with a non smooth zero level). The distance we use between polynomials is induced by the Bombieri norm [1] for which we establish a nice formula for the distance to the discriminant which is the main tool of our proof. 1. Notation Let S n−1 be the unit sphere of R n. In R n, we write ‖x ‖ the usual Euclidean norm. We consider E = R[X1,...,Xn]d the vector space of homogeneous polynomials in n> 1 variables of degree d> 1. Let N be the dimension of this vector space, we have N = () d+n−1 n−1 ≥ n Let 〈 , 〉 be a scalar product on E and ‖ ‖ the associated norm. We use the same notation for the scalar product and the norm of E and the euclidian one’s on R n, the context should make it clear what norm we are using. Let B = (E1,...,EN) be an orthonormal basis of E. For x ∈ Rn, C(x) denotes the line vector (E1(x),...,EN(x)) and Bi(x) for i ∈ {1,...,n} denotes the line vector ( ∂E1(x)