Let $X\subset {ℝ}^n$ be a compact subanalytic surface. This paper shows that, in a suitable sense, there are very few rational points of $X$ that do not lie on some connected semialgebraic curve contained in $X$.

This paper presents several theorems on the rectilinearization of functions definable by a convergent Weierstrass system, as well as their applications to decomposition into special cubes and quantifier elimination.

Neither real analytic sets nor the images of real or complex analytic mappings are, in general, coherent. Let $\Phi :X\to Y$ be a morphism of real analytic spaces, and let $\Psi :𝒢\to ℱ$ be a homomorphism of coherent modules over the induced ring homomorphism ${\Phi }^{*}:{𝒪}_Y\to {𝒪}_X$. We conjecture that, despite the failure of coherence, certain natural discrete invariants of the modules of formal relations ${ℛ}_a=\mathrm{Ker}{\widehat{\Psi }}_a$, $a\in X$, are upper semi-continuous in the analytic Zariski topology of $X$. We prove semicontinuity in many cases (e.g. in the algebraic category)....

This is a sequel to “Relations among analytic functions I”, Ann. Inst. Fourier, 37, fasc. 1, [pp. 187-239]. We reduce to semicontinuity of local invariants the problem of finding ${𝒞}^{\infty }$ solutions to systems of equations involving division and composition by analytic functions. We prove semicontinuity in several general cases : in the algebraic category, for “regular” mappings, and for module homomorphisms over a finite mapping.

Given a real analytic manifold Y, denote by $Y_{sa}$ the associated subanalytic site. Now consider a product Y = X × S. We construct the endofunctor $ℱ\mapsto {ℱ}^S$ on the category of sheaves on $Y_{sa}$ and study its properties. Roughly speaking, ${ℱ}^S$ is a sheaf on $X_{sa}\times S$. As an application, one can now define sheaves of functions on Y which are tempered or Whitney in the relative sense, that is, only with respect to X.