dr. Lerario's seminar at SISSA

[Emanuele Tuillier Illingworth]

SEMINAR ANNOUNCEMENT


Wednesday Feb. 26, 16:00, room 133.

A. Lerario (Institut Camille Jordan, Lyon)

"Counting the number of nodal lines of a random polynomial"

To determine the average number of real zeroes of a univariate 
polynomial whose coefficients are random variables is a classical problem.

A natural way to generalize it is to ask for the average number of 
connected components of the zero set of a random polynomial in several 
variables. This question is much influenced by a random approach to 
Hilbert's Sixteenth Problem, to study the number and the arrangement in 
the projective space of the components of a real algebraic hypersurface.

The answer to the above questions (both in the univariate and the 
multivariable case) strongly depends on the choice of the probability 
distribution.

In this talk I will show that in fact the case of several variables can 
be reduced to the classical univariate problem - using a sort of average 
hyperplane section theorem.

More precisely, the number of connected components of a random 
hypersurface in RP^n of degree d has the same order of the number of 
points of intersection of this hypersurface with a fixed projective 
line, raised to the n-th power.

The methods combine harmonic analysis and random matrix theory, using a 
distribution result for the Morse index of the critical points of a 
random function on the sphere.

(This is joint work with Y. Fyodorov and E. Lundberg)