Distance to the discriminant
Résumé
We will study algebraic hyper-surfaces on the real unit sphere 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 work about the topology of random hyper-surfaces [4, 3]. Our first result (lemma 2.2 page 4) 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 prop- erties. For instance we establish a combinatoric formula for the scalar product of two products of linear-forms (lemma 3.2 page 5) 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 4.3 page 9 from which we deduce the following inequality: dist(P , ∆) ≤ min_{x critical point of P on S^{n−1}} |P (x)| The definition 4.2 page 8 classifies in two categories the ways to make a polynomial singular to realise the distance to the discriminant. Then, we show, in theorem 5.3 page 15, that one of the category is forbidden in the case of an extremal hyper-surfaces (i.e. with maximal Betti numbers). This implies as a corollary 5.4 (page 19) 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 ob jects which enjoy properties similar to those of quadratic forms: they are linear combination of power of linear forms x → (x|u_i)^d where the vector u_i are the critical points of P on S^{n−1} corresponding to the least positive critical value of |P|. This is corollary 6.2 page 21 of a similar theorem 6.1 page 20 for all algebraic hyper-surfaces. We also obtain metric information about algebraic hyper-surfaces. First, in the case of extremal hyper-surface, we give an upper bound (theorem 7.3 page 22) 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 (corollary 8.1 and theorem 8.2). 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).
Domaines
Géométrie algébrique [math.AG]
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