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

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, for example, $Maps((k^n,o),(k^m,o))$, are usually studied up to the left–right equivalence. The natural question is the left–right version of AP: when is the formal left–right equivalence of morphisms approximated by the “ordinary” (i.e., analytic, resp., algebraic) equivalence? In this case, the standard AP is not directly applicable, as the involved (functional) equations are not of implicit function type. Moreover, the naive extension does not hold in the analytic case, because of Osgood–Gabrielov–Shiota examples. The left–right version of Artin approximation ($LR$.AP) was established by M. Shiota for morphisms that are either Nash or [real-analytic and of finite singularity type]. We establish $LR$.AP and its stronger version of Płoski ($LR$.APP) for $Maps(X,Y)$, where $X,Y$ are analytic/algebraic germs of schemes of any characteristic. More precisely:

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 nonpathological 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.