As it is well-known, the steepness property is a local geometric transversality condition on the gradient of C2
functions which proves fundamental in order to ensure the stability over long timespans of integrable Hamiltonian systems that undergo a small perturbation. Though steepness is generic - both in measure and in topological sense - among functions of high enough regularity, the original definition of this property is not constructive and, up to very recent times, the few existing criteria to check steepness were non-generically verified and applied only to functions depending on a low number of variables. By combining Yomdin's Lemma on the analytic reparametrization of semi-algebraic sets together with non-trivial estimates on the codimension of suitable real-algebraic varieties, in this talk I will state explicit algebraic criteria for steepness which are generically verified and apply to functions depending on any number of variables. This constitutes a very important result for applications, e.g. in celestial mechanics. The criteria can be constructed recursively and are based on algebraic equalities involving the derivatives of the studied function up to any given order and external real parameters, some of which belong to compact sets and some others to non-compact sets. Moreover, it can be shown that, generically, the non-compact external parameters can be eliminated from the equalities with the help of a linear quantifier elimination algorithm: this represents a crucial improvement for numerical implementations of the criteria.
References:
1) S. Barbieri, "Semi-algebraic Geometry and generic Hamiltonian stability", preprint. https://hal.science/hal-04213250/
2) S. Barbieri, "On the algebraic properties of exponentially stable integrable hamiltonian systems", Ann. Fac. Sci. Toulouse, 31(6): 1365-1390, 2022
3) N. N. Nekhoroshev, "Stable lower estimates for smooth mappings and for gradients of smooth functions", Math USSR Sb., 19(3):425–467, 1973
4) G. Schirinzi, M. Guzzo, "On the formulation of new explicit conditions for steepness from a former result of N.N. Nekhoroshev", J. Math. Phys, 54, 2013
