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Degenerations of classical square matrices and their determinantal structureMedeiros, Rainelly Cunha de 10 March 2017 (has links)
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Previous issue date: 2017-03-10 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In thisthesis,westudycertaindegenerations/specializationsofthegenericsquare
matrix overa eld k of characteristiczeroalongitsmainrelatedstructures,suchthe
determinantofthematrix,theidealgeneratedbyitspartialderivatives,thepolarmap
de ned bythesederivatives,theHessianmatrixandtheidealofsubmaximalminorsof
the matrix.Thedegenerationtypesofthegenericsquarematrixconsideredhereare:
(1) degenerationby\cloning"(repeating)avariable;(2)replacingasubsetofentriesby
zeros, inastrategiclayout;(3)furtherdegenerationsoftheabovetypesstartingfrom
certain specializationsofthegenericsquarematrix,suchasthegenericsymmetric
matrix andthegenericsquareHankelmatrix.Thefocusinallthesedegenerations
is intheinvariantsdescribedabove,highlightingonthehomaloidalbehaviorofthe
determinantofthematrix.Forthis,weemploytoolscomingfromcommutativealgebra,
with emphasisonidealtheoryandsyzygytheory. / Nesta tese,estudamoscertasdegenera c~oes/especializa c~oesdamatrizquadradagen erica
sobre umcorpo k de caracter sticazero,aolongodesuasprincipaisestruturasrela-
cionadas, taiscomoodeterminantedamatriz,oidealgeradoporsuasderivadasparci-
ais, omapapolarde nidoporessasderivadas,amatrizHessianaeoidealdosmenores
subm aximosdamatriz.Ostiposdedegenera c~aodamatrizquadradagen ericacon-
siderados aquis~ao:(1)degenera c~aopor\clonagem"(repeti c~ao)deumavari avel;(2)
substitui c~aodeumsubconjuntodeentradasporzeros,emumadisposi c~aoestrat egica;
(3) outrasdegenera c~oesdostiposacimapartindodecertasespecializa c~oesdamatriz
quadrada gen erica,taiscomoamatrizgen ericasim etricaeamatrizquadradagen erica
de Hankel.Ofocoemtodasessasdegenera c~oes enosinvariantesdescritosacima,
com destaqueparaocomportamentohomaloidaldodeterminantedamatriz.Paratal,
empregamos ferramentasprovenientesda algebracomutativa,com^enfasenateoriade
ideais enateoriadesiz gias.
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