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The classical Artin approximation (AP) reads: any formal solution of a systemof (analytic, resp. algebraic) equations of implicit function type isapproximated by ``ordinary" solutions (i.e. analytic, resp. algebraic).Morphisms of scheme-germs, e.g. Maps((k^n,o),(k^m,o)) are usually studied up tothe left-right equivalence. The natural question is the left-right version ofArtin approximation: when is the formal left-right equivalence of morphismsapproximated by the ``ordinary" (i.e. analytic, resp. algebraic) equivalence?In this case the standard Artin approximation is not directly applicable, asthe involved (functional) equations are not of implicit function type.Moreover, the na\"ive extension does not hold in the analytic case, because ofOsgood-Gabrielov-Shiota examples. The left-right version of Artin approximation(LRAP) was established by M. Shiota for morphisms that are either Nash or[real-analytic and of finite singularity type]. We establish LRAP and itsstronger version of P\l oski (LRAPP) for Maps(X,Y) where X,Y areanalytic/algebraic germs of schemes of any characteristic. More precisely: *LRAP, LRAPP, the inverse Artin approximation (and its P\l oski's version) holdfor algebraic morphisms and for finite analytic morphisms. * LRAP holds foranalytic morphisms of weakly-finite singularity type. (For char>0 we imposecertain integrability condition.)  This latter class of morphisms of ``weakly-finite singularity type" (which weintroduce) is of separate importance. It extends naturally the traditionalclass of morphisms of ``finite singularity type", while preserving theirnon-pathological behavior. The definition goes via the higher critical loci andhigher discriminants of morphisms with singular targets.  We establish basic properties of these critical loci. In particular: any mapis finitely (right) determined by its higher critical loci.

Results on left-right approximation for algebraic morphisms and for analytic morphisms of weakly finite singularity type