|
Abstract: The classical Bezout theorem implies an explicit upper bound on the
number of isolated zeros (either real or complex) of algebraic functions
and their polynomial combinations in terms of the degrees involved. In
general, one cannot count so easily zeros of arbitrary analytic
functions. However, for functions satisfying polynomial ordinary
differential equations, a "restricted quasialgebraicity" holds: zeros of
such functions and their polynomial combination can be counted in
bounded subdomains of the real line or the complex plane. Ultimately,
for solutions of Fuchsian (linear ordinary differential) equations
having only simple poles of the coefficients, the "global
quasialgebraicity" can be proved and bounds for zeros on the whole
Riemann sphere obtained in terms of norms of the residue matrices.
If time permits, we discuss connections between these results and the
tangential Hilbert 16th problem on limit cycles appearing by
perturbations of planar Hamiltonian polynomial systems. (Joint work with Dmitry Novikov)
|
