Distance to the Discriminant

We will study algebraic hyper-surfaces on the real unit sphere Sn−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. This view point is mostly present in works about the topology of random hyper-surfaces [5, 4]. Our first result (lemma 3.2 page 5) is a formula for the distance dist(P,∆) of a polynomial to the real discriminant ∆, i.e. the set of polynomials with a real singularity on the sphere. This formula is given for any distance coming from a scalar product on the vector space of polynomials. Then, we concentrate on Bombieri scalar product and its remarkable properties. For instance we establish a combinatoric formula for the scalar product of two products of linear-forms (lemma 4.2 page 6) which allows 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 in theorem 5.3 page 9 from which we deduce the following inequality: