### Abstract

The application of the Nekhoroshev theorem to many problems arising in different fields of Physics and Astronomy depends on a non-degeneracy property, called steepness, that a suitable Hamiltonian approximation must satisfy. Since steepness is implicitly defined, we have the problem of recognizing whether a given function is steep or not. For this purpose, we here consider some sufficient conditions for steepness provided by Nekhoroshev in 1979, based on the solvability of a collection of systems depending on the number n of degrees of freedom, the derivatives of the function up to a certain order r, and some auxiliary parameters. These conditions are really explicit only for r = 2, corresponding to quasi-convexity , and for r = 3. Instead, for r ⩾ 4, the conditions are implicit, since they require an elaborate computation of the closure of a certain set. In this paper, we first revisit Nekhoroshev's result and we show that the number of parameters in the collections of systems can be suitably reduced. Then, we show that for r = 4 Nekhoroshev's result is interesting only for n = 2, 3, and 4, and in these cases we find explicit conditions for steepness which are formulated in a purely algebraic form.

Received 11 February 2013
Accepted 17 June 2013
Published online 15 July 2013

Acknowledgments:
We wish to thank G. Benettin for many useful discussions about steepness (and also for example 1 in Sec. II).

Article outline:

I. INTRODUCTION
II. NEW EXPLICIT CONDITIONS FOR STEEPNESS
III. REVIEW OF SCIENTIFIC CONDITIONS FOR STEEPNESS FORMULATED BY NEKHOROSHEV
IV. EXPLICIT EXPRESSION FOR THE SYSTEMS DEFINING σ^{ r }(*n*)
A. The special case *m* = 1
B. The special cases β_{1} = 2 and β_{1} = 3
V. PROOFS OF PROPOSITIONS 2.2, 2.3, AND 2.4
A. Proof of Proposition 2.2: Steepness along the 4-jet of functions of two variables
B. Proof of Proposition 2.3: Steepness along the 4-jet of functions of three variables
1. Explicit formulation of σ^{4}(3)
2. A reduced formulation of **S** _{ 2 }(*h*)
3. A closed set containing σ^{4}(3)
C. Proof of Proposition 2.4: Steepness along the 4-jet of functions of four variables
1. Explicit formulation of σ^{4}(4)
2. A reduced formulation of **S** _{ 2 }(*h*) and **S** _{ 3 }(*h*)
3. A closed set containing σ^{4}(4)

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