Regular chain - Wikipedia

In computer algebra, a regular chain is a particular kind of triangular set in a multivariate polynomial ring over a field. It enhances the notion of ... Regularchain FromWikipedia,thefreeencyclopedia Jumptonavigation Jumptosearch Incomputeralgebra,aregularchainisaparticularkindoftriangularsetinamultivariatepolynomialringoverafield.Itenhancesthenotionofcharacteristicset. Contents 1Introduction 2Examples 3Formaldefinitions 4Properties 5Seealso 6Furtherreferences Introduction[edit] Givenalinearsystem,onecanconvertittoatriangularsystemviaGaussianelimination.Forthenon-linearcase,givenapolynomialsystemFoverafield,onecanconvert(decomposeortriangularize)ittoafinitesetoftriangularsets,inthesensethatthealgebraicvarietyV(F)isdescribedbythesetriangularsets. Atriangularsetmaymerelydescribetheemptyset.Tofixthisdegeneratedcase,thenotionofregularchainwasintroduced,independentlybyKalkbrener(1993),YangandZhang(1994).RegularchainsalsoappearinChouandGao(1992).Regularchainsarespecialtriangularsetswhichareusedindifferentalgorithmsforcomputingunmixed-dimensionaldecompositionsofalgebraicvarieties.Withoutusingfactorization,thesedecompositionshavebetterpropertiesthattheonesproducedbyWu'salgorithm.Kalkbrener'soriginaldefinitionwasbasedonthefollowingobservation:everyirreduciblevarietyisuniquelydeterminedbyoneofitsgenericpointsandvarietiescanberepresentedbydescribingthegenericpointsoftheirirreduciblecomponents.Thesegenericpointsaregivenbyregularchains. Examples[edit] DenoteQtherationalnumberfield.InQ[x1,x2,x3]withvariableorderingx1