Distance to the discriminant

TL;DR

This paper investigates the geometric properties of algebraic hyper-surfaces on the sphere, deriving formulas for the distance to the discriminant set using Bombieri's scalar product, and characterizes extremal hyper-surfaces with maximal Betti numbers.

Contribution

It provides a new formula for the distance to the discriminant using Bombieri's scalar product and characterizes extremal hyper-surfaces that maximize this distance.

Findings

Derived a formula for the distance to the discriminant for any scalar product.

Established the invariance of Bombieri's norm under orthogonal transformations.

Proved that extremal hyper-surfaces with maximal Betti numbers attain the maximum distance to the discriminant.

Abstract

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 identities that we could not find somewhere else (proposition \ref{trigo} page \pageref{trigo}). We also obtain metric information about algebraic hyper-surfaces. First, in the case of extremal hyper-surface, we give an upper bound (theorem \ref{bandwidth} page \pageref{bandwidth}) on the length of an integral curve of the gradient of $P$ in the band where $|P|$ is less that the least positive critical value of $|P|$. Then, a general lower bound on the size and distance between the connected components of the zero locus of $P$ (corollary \ref{sphereinside} and theorem \ref{distancebetween}). The last section will present experimental results among which are five extremal sextic curves far from the discriminant. These are obtained by very long running numerical optimisation (many months) some of which are not terminated.