Distance to the discriminant

Christophe Raffalli (1)

1 LAMA - Laboratoire de Mathématiques

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 $\mathcal P^n(\mathbb R)$ 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 for which we establish a nice formula for the distance to the discriminant which is the main tool of our proof.

Domaines: Géométrie algébrique [math.AG]

Christophe Raffalli. Distance to the discriminant. 2008. ⟨hal-00396984v1⟩