Seminario de Álgebra IMERL

Seminario de Álgebra IMERL
Distance to the discriminant for real polynomials.
02/06/2017 de 11:00 a 12:00 Salón de seminarios - IMERL,

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…