English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

The standard Artin approximation (AP) is not readily applicable to theleft-right equivalence of map-germs Maps((k^n,o),(k^m,o)). Moreover, the naiveextension does not hold in the k-analytic case, because ofOsgood-Gabrielov-Shiota examples.  The left-right version of Artin approximation (LRAP) was established by M.Shiota for map-germs that are either Nash or [real-analytic and of finitesingularity type].  We establish LRAP for Maps(X,Y) where X,Y are k-analytic/k-Nash germs ofschemes of any characteristic. More precisely: * (k-Nash power series, k<x>/J_Xand k<y>/J_Y.) LRAP holds for any map. * (k-analytic power series, k{x}/J_X andk{y}/J_Y.) LRAP holds for maps of weakly-finite singularity type.  This ``weakly-finite singularity type" (which we introduce) is a naturalextension of the classical finite singularity type, and is of separateinterest.  As a trivial corollary we get: the inverse Artin approximation holds foranalytic maps of weakly-finite singularity type.  Then we extend the properties RAP, LAP, LRAP to the approximation results forquivers of maps, Gamma-AP.  Finally, we establish the nested version, ``Gamma-AP with parameters". It isneeded for families/unfoldings of maps.

Artin approximation for left-right equivalence of map-germs and for quivers of map-germs