**abstract:**
The notion of steepness for a given function was firstly
introduced by N.N. Nekhoroshev in the Seventies. A function which is
real-analytic around a compact set of $\mathbb{R}^{n$} is steep if and
only if it has no critical points and its restriction to any affine
hyperplane of $\mathbb{R}^{n$} admits only isolated critical points
(Niederman, 2006). Such property is related to the stability of
finite-dimensional hamiltonian systems which are close to integrable,
i.e. whose hamiltonian function is the sum of an integrable part and a
small perturbation. By making use of canonical action-angle
coordinates to describe such system, it is well known that the actions
stay constant under the integrable flow, whereas diffusion is possible
when a perturbation is added, provided the system has three or more
degrees of freedom. A celebrated result by Nekhoroshev (1977) states
that, nevertheless, the action variation is small over a time which is
exponential in the inverse of the size of the perturbation, provided
that the integrable part of the hamiltonian is steep. In this light,
understanding wether a given function is steep or not turns out to be
a crucial issue. Unfortunately, the simple definition of steepness is
not easy to verify directly. However, in his work of 1979, Nekhoroshev
provided the way to explicitly construct a semi-algebraic set in the
space of r-jets of functions (that is, the space of the Taylor
polynomials calculated at a certain point x, up to a fixed order r\geq
2) whose closure contains the jets of all non-steep functions with
non-zero gradient at x. Indeed, a function whose r-jet at x is
contained in the complementary of the closure of such semi-algebraic
set is steep at x. The explicit computations involved in this
construction have been explicitly carried out by Nekhoroshev (1977)
for r=2,3 and by Schirinzi and Guzzo (2013) for r=4. In this seminar,
I will show the construction for r=5 and, therefore, provide new
explicit algebraic conditions for steepness on the coefficients of the
Taylor polynomial of order five of a given smooth function. Besides
loosening the sufficient conditions for a function to be steep, this
works also allows for a deeper insight in the construction of
sufficient conditions at any order r of the jet and this, in turn,
leads to the formulation of interesting conjectures

Thu 28 Mar, 14:30 - 15:30, Sala Conferenze Centro De Giorgi

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