[Todos CMAT] Charla del Prof. Cristophe Raffalli

[Fernando Abadie]

Estimados:

El Profesor Cristophe Raffalli, de la Université de Savoie, dará este
*viernes 9 de junio* una charla a la que los invito a asistir, y cuyos
título y resumen incluyo más abajo.
Será en el CMAT de la Facultad de Ciencias, en el *salón de*
*seminarios del piso 16, a las 10 horas*.


Saludos,
Fernando



*Distance to the discriminant for real polynomials.*

The question of the possible topologies of real algebraic
hypersurfaces in the projective space of dimension n is still open
(16th Hilbert's problem). For instance, the maximum number of
connected components (b_0) of a surface of degree 5 is still unknown
(we know that it is 23, 24 or 25). The complete classification of
curves of degree 8 is also not finished. There are now very few
results (none since 2001 ?) and new directions are necessary.

In this talk, we adopt an Euclidian point of view using the
Kostlan/Bombieri norm that we will introduce. We give a simple
expression for the distance d(P,Δ) between a polynomial P and the real
discriminant Δ (the set of real polynomials with a real
singularity). Moreover, for hypersurfaces with a "locally extremal"
topology, d(P,Δ) is equal to the least non zero critical value of |P|
on the unit sphere of Rⁿ. We will define this notion of "locally
extramal", but it includes hypersurfaces with maximum sum of Betti
numbers for a given degree and also the empty ones.

In this particular case, polynomials that maximise the distance to the
discriminant, with a fixed degree and norm, can be written as sums of
d-forms (i.e. L(x) = (x.u)ᵈ) where d is the degree of the polynomial.
Moreover, the directions of the linear forms (the u above) are the
points where the least critical value of |P| is reached.

We identify exactly this special polynomials in two cases:

- homogeneous polynomial in 2 variables with the maximum number of
  real roots.

- positive polynomials in any number of variables or degree.

In each cases, we get an interesting corollary...

If enough time, we could discuss the number of terms in the sums of
d-forms mentioned above and show some sharp lower-bounds for some
curves of degree 6 (Those that interested Hilbert when he stated his
16th problem).

Among the numerous open questions introduced in the work, we could
hope (or dream?), to find a way to have both sharp lower and upper
bounds for this number of terms and deduce that an unresolved
topological type is in fact impossible because the bounds are
incompatible. However, the problem of upper bounds seems very hard…
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