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Distance to the discriminant

Le : 07/11/2014 14h00
Par : Christophe Raffalli (Chambéry)
Lieu : I 001
Lien web :
Résumé : We will study algebraic hyper-surfaces on the real unit sphere S of dimension n-1 (given by an homogeneous polynomial of degree d in n variables) with the view point rarely exploited of Euclidian geometry using Bombieri's scalar product and norm. We will first show some remarkable properties of this scalar product, for instance a combinatoric formula for the scalar product of two products of linear-forms which allow to give a (new ?) proof of the invariance of Bombieri's norm by composition with the orthogonal group. These properties yield a simple formula for the distance of an algebraic hyper-surface to the "real discriminant" (the set of hyper-surfaces with a real singularity on the sphere). This property can be further simplified when the hyper-surface has extremal Betti numbers. In this case we have dist({x in S |P(x)=0}, Delta) = min_{x critical point of P on S} |P(x)| Finally, we will show that extremal hyper-surfaces that maximize the distance to the discriminant are very remarkable objects. We will illustrate the talk showing all extremal sextics curves far from the discriminant and obtained by numerical optimisation (hence no waranty).