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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”, , , 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.