Results on left-right approximation for algebraic morphisms and for analytic morphisms of weakly finite singularity type
Dmitry Kerner
TL;DR
This work advances left-right Artin approximation (LRAP) and its parameterized variant (LRAPP) for morphisms between scheme-germs beyond the real-analytic Nash setting, covering algebraic and finite analytic cases and introducing a broader class called weakly-finite singularity type (w.f.s.t.). Central to the development is the construction and analysis of higher critical loci and higher discriminants, which yield finite determinacy results: every dominant map f is finitely determined by CritX f and by Criti for suitable i. The authors show LRAP, LRAPP, and inverse Artin-type results for algebraic morphisms and for finite analytic morphisms, and prove LRAP for analytic morphisms of w.f.s.t. (with extra integrability/lifting hypotheses in positive characteristic). The methodology hinges on recasting LRAP as a base-change problem between algebras, deploying nested Artin/Ploski approximation, and leveraging approximate integrability and lifting of automorphisms to propagate local equivalences to global ones, with a strong emphasis on the geometry of the critical loci. The results unify and extend Shiota’s left-right approximation framework to a broad algebraic and singular setting, offering a robust toolkit for studying deformations and unfoldings of morphisms across arbitrary characteristics. The work provides foundational definitions, computational strategies, and determinacy principles that have potential impacts on singularity theory, deformation theory, and moduli problems in algebraic geometry.
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 naïve 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 Płoski (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 Płoski'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.