On the algebraic properties of exponentially stable integrable hamiltonian systems

Steepness is a generic geometric property which, together with complex-analyticity, needed in order to ensure stability of nearly-integrable hamiltonian system over exponentially long times. Following strategy developed by Nekhoroshev, we construct sufficient conditions for steepness given function that involve algebraic equations on its derivatives up five. This important view applications (e.g. Celestial Mechanics). The underlying analysis suggests some interesting considerations the genericity steepness. Moreover, this work represents first step towards construction involving studied an arbitrary order.