polynomials and polynomial inequalities

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1、ForTheresa,Sophie,Alexandra,ennifer,andErikarefa eolynomialspervademathemati s,andmu hthatisbeautifulinmathe-mati sisrelatedtopolynomials.Virtuallyeverybran hofmathemati s,fromalgebrai numbertheoryandalgebrai geometrytoappliedanaly-sis,Fourieranalysis,and

2、 omputers ien e,hasits orpusoftheoryarisingfromthestudyofpolynomials.Histori ally,questionsrelatingtopolyno-mials,forexample,thesolutionofpolynomialequations,gaverisetosomeofthemostimportantproblemsoftheday.Thesubje tisnowmu htoolargetoattemptanen y lopedi

3、overage.Thebodyofmaterialwehoosetoexplore on ernsprimarilypolyno-mialsastheyariseinanalysis,andthete hniquesofthebookareprimarilyanalyti .Whilethe onne tingthreadisthepolynomial,thisisananalysisbook.Thepolynomialsandrationalfun tionsweare on ernedwitharealmo

4、stex lusivelyofasinglevariable.Weassumeatmostaseniorundergraduatefamiliaritywithrealand omplexanalysis(indeedinmostpla esmu hlessisrequired).However,thematerialisoftenterselypresented,withmu hmathemati sexploredintheexer ises,someofwhi harequitehard,manyofwh

5、i haresuppliedwith opioushints,somewith ompleteproofs.Welloverhalfthematerialinthebookispresentedintheexer ises.Thereaderisen ouragedtoatleastbrowsethroughthese.Wehavebeenmu hin uen edbyolyaandSzeg}o's lassi roblemsandTheoremsinAnalysis"inourapproa htoth

6、eexer ises.(ThoughunlikeolyaandSzeg}owehosetoin orporatethehintswiththeexer ises.)viiirefa eThebookismostlyself- ontained.Thetext,withouttheexer ises,pro-videsanintrodu tiontothematerial,butmu hoftheri hnessisreservedfortheexer ises.Wehaveattemptedtohighl

7、ightthepartsofthetheoryandthete hniqueswendmostattra tive.So,forexample,untz'slovelyhara terizationofwhenthespanofasetofmonomialsisdenseisexploredinsomedetail.Thisresultepitomizesthebestofthesubje t:anattra tiveandnontrivialresultwithseveralattra tiveandno

8、ntrivialproofs.Thereareex ellentbooksonorthogonalpolynomials,Chebyshevpoly-nomials,Chebyshevsystems,andthegeometryofpolynomials,tonamebutafewofthetopi swe over,anditisnotourintenttorewriteanyoft