[Todos CMAT] Seminario de álgebra del IMERL: Christophe Raffalli

[Ana González]

Estimados,
este viernes vamos a contar con la participación de Christophe Raffalli,
del laboratorio LAMA, UMR 5127 CNRS de la Université Savoie Mont Blanc. A
continuación se detalla título y resumen de la charla.
Les recuerdo el seminario comienza a las 11:!5 y es en el salón de
seminarios del IMERL.

*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… *

Saludos cordiales.
Ana González
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