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

Abstract

The classical Artin approximation (AP) reads: any formal solution of a system of (analytic, resp. algebraic) equations of implicit function type is approximated 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 to the left-right equivalence. The natural question is the left-right version of Artin approximation: when is the formal left-right equivalence of morphisms approximated by the ``ordinary" (i.e. analytic, resp. algebraic) equivalence? In this case the standard Artin approximation is not directly applicable, as the involved (functional) equations are not of implicit function type. Moreover, the na\"ive extension does not hold in the analytic case, because of Osgood-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 its stronger version of P\l oski (LRAPP) for Maps(X,Y) where X,Y are analytic/algebraic germs of schemes of any characteristic. More precisely: * LRAP, LRAPP, the inverse Artin approximation (and its P\l oski's version) hold for algebraic morphisms and for finite analytic morphisms. * LRAP holds for analytic morphisms of weakly-finite singularity type. (For char>0 we impose certain integrability condition.) This latter class of morphisms of ``weakly-finite singularity type" (which we introduce) is of separate importance. It extends naturally the traditional class of morphisms of ``finite singularity type", while preserving their non-pathological behavior. The definition goes via the higher critical loci and higher discriminants of morphisms with singular targets. We establish basic properties of these critical loci. In particular: any map is finitely (right) determined by its higher critical loci.