On the pythagoras numbers of real analytic surfaces

Francesca Acquistapace; Fabrizio Broglia; José F. Fernando; Jesús M. Ruiz

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

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