Distance to the discriminant
  We will study algebraic hyper-surfaces on the real unit sphere $\mathcal S^{n-1}$ 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. This view point is mostly present in works about the topology of random hyper-surfaces \cite{ShubSmale93, GayetWelschinger14}. Our first result (lemma \ref{distgen} page \pageref{distgen}) is a formula for the distance $\dist(P,\Delta)$ of a polynomial to the {\em real discriminant} $\Delta$, 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 \ref{scalarlinear} page \pageref{scalarlinear}) 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 \ref{bombineqdist} page \pageref{bombineqdist} from which we deduce the following inequality: $$\dist(P, \Delta) \leq \min_{x \hbox{ critical point of } P \hbox{ on } \mathcal S^{n-1}} |P(x)|$$ The definition \ref{maindef} page \pageref{maindef} classifies in two categories the ways to make a polynomial singular to realise the distance to the discriminant. Then, we show, in theorem \ref{extremal} page \pageref{extremal}, that one of the category is forbidden in the case of an {\em extremal} hyper-surfaces (i.e. with maximal Betti numbers). This implies as a corollary \ref{bombeqdistbis} (page \pageref{bombeqdistbis}) that the above inequality becomes an equality is that case. The main result in this paper concerns extremal hyper-surfaces $P=0$ that maximise the distance to the discriminant (with $\|P\| = 1$). They are very remarkable objects which enjoy properties similar to those of quadratic forms: they are linear combination of powers of linear forms $x \mapsto \langle x | u_i \rangle^d$ where the vectors $u_i$ are the critical points of $P$ on $\mathcal S^{n-1}$ corresponding to the least positive critical value of $|P|$. This is corollary \ref{comblinbis} page \pageref{comblinbis} of a similar theorem \ref{comblin} page \pageref{comblin} for all algebraic hyper-surfaces. The next section is devoted to homogeneous polynomials in $2$ variables. We prove that a polynomial of degree $d$ with $2d$ regularly spaced roots on the unit circle is a local maximum of the distance to the discriminant among polynomials with the same norm and number of roots. We conjecture that it is a global maximum and that the polynomial of degree $d$ with $2r$ regularly spaced roots on the unit circle is also a similar global maximum when $d < r \leq 2d$. This claim is supported by the fact that we were able to prove the consequence of this together with corollary \ref{comblinbis} which yields to interesting trigonometric...
Christophe Raffalli (LAMA)
math.AG