  The steepness property is a local geometric transversality condition on the
gradient of a $C^2$-function which is fundamental in order to ensure the
stability of sufficiently-regular nearly-integrable Hamiltonian systems over
long timespans. Steep functions were originally introduced by Nekhoroshev, who
also proved their genericity. Namely, given a pair of positive integers $r,n$,
with $r$ high enough, and a point $I_0\in \mathbb{R}^n$, the Taylor polynomials
of those $C^{2r-1}$ functions which are not steep around $I_0$ are contained in
a semi-algebraic set of positive codimension in the space of polynomials of $n$
variables and degree bounded by $r$. The demonstration of this result was
originally published in 1973 and has been hardly studied ever since, probably
due to the fact that it involves no arguments of dynamical systems: it makes
use of quantitative reasonings of real-algebraic geometry and complex analysis.
The aim of the present work is two-fold. In the first part, the original proof
of the genericity of steepness is rewritten by making use of modern tools of
real-algebraic geometry: this allows to clarify the original reasonings, that
were obscure or sketchy in many parts. In particular, Yomdin's Lemma on the
analytic reparametrization of semi-algebraic sets, together with non trivial
estimates on the codimension of certain algebraic varieties, turn out to be the
fundamental ingredients to prove the genericity of steepness. The second part
of this work is entirely new and is devoted to the formulation of explicit
algebraic criteria to check steepness of any given sufficiently regular
function, which constitutes a very important result for applications, as the
original definition of steepness is not constructive. These criteria involve
both the derivatives of the studied function up to any given order and external
real parameters that, generically, belong to compact sets.
