We prove the o-minimal generalization of the Łojasiewicz inequality $\parallel \mathrm{grad}f\parallel \ge |f{|}^{\alpha }$, with $\alpha \<1$, in a neighborhood of $a$, where $f$ is real analytic at $a$ and $f\left(a\right)=0$. We deduce, as in the analytic case, that trajectories of the gradient of a function definable in an o-minimal structure are of uniformly bounded length. We obtain also that the gradient flow gives a retraction onto levels of such functions.

Let R be a real closed field, and denote by ${}_{R,n}$ the ring of germs, at the origin of Rⁿ, of $C^{\infty }$ functions in a neighborhood of 0 ∈ Rⁿ. For each n ∈ ℕ, we construct a quasianalytic subring ${}_{R,n}{\subset }_{R,n}$ with some natural properties. We prove that, for each n ∈ ℕ, ${}_{R,n}$ is a noetherian ring and if R = ℝ (the field of real numbers), then ${}_{ℝ,n}=ₙ$, where ₙ is the ring of germs, at the origin of ℝⁿ, of real analytic functions. Finally, we prove the Real Nullstellensatz and solve Hilbert’s 17th Problem for the ring ${}_{R,n}$.

The paper establishes the basic algebraic theory for the Gevrey rings. We prove the Hensel lemma, the Artin approximation theorem and the Weierstrass-Hironaka division theorem for them. We introduce a family of norms and we look at them as a family of analytic functions defined on some semialgebraic sets. This allows us to study the analytic and algebraic properties of this rings.