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The Fundamental Theorem of Algebra made effective: an elementary real algebraic proof via Sturm chains

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The Fundamental Theorem of Algebra made effective: an elementary real-algebraic proof via Sturm chains Michael Eisermann Institut Fourier, Universite Grenoble www-fourier.ujf-grenoble.fr/˜eiserm January 6, 2009 Carl Friedrich Gauss (1777–1855) Augustin Louis Cauchy (1789–1857) Charles-Franc¸ois Sturm (1803–1855) MAA–AMS Joint Mathematics Meetings in Washington DC AMS Session on Analytic Function Theory

  • exist z1

  • complex polynomial

  • ujf grenoble

  • real numbers

  • appealing proof

  • algebra

  • algebra made effective

  • there exist


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Language English
The Fundamental Theorem of Algebra made effective: an elementary real-algebraic proof via Sturm chains
Carl Friedrich Gauss (1777–1855)
Michael Eisermann
Institut Fourier, Universit ´e Grenoble www-fourier.ujf-grenoble.fr/˜eiserm
January 6, 2009
Augustin Louis Cauchy (1789–1857)
Charles-Franc¸ ois Sturm (1803–1855)
MAA–AMS Joint Mathematics Meetings in Washington DC AMS Session on Analytic Function Theory
plicreexMo=RtClend=i2],[ieroehT.1reveroFm:LetitlyheeRbeterladlforeasunbm0w+ahcitplomcoexcfetnei,0as.,1aypolynomialF=Zn+na1nZ1+a+Z11zZ(=FtahthcusCn)z(Z)z2Z)(erxetCehn1..a,,zn,...1,z2istzmoegirteratnteyylieaprngllcappyauqseitnoN.tarulaeanelemes:IstherewtsC?nahtneergnordehicheldsredhtopyhehwot?sisenwCaf?oontkeeaew
of
theorem
fundamental
The
Theorem
complex
Every
polynomial
of
degree
n
has
n
complex
roots.
tiekeffevitc
algebra
?eecthcloniousman?
veryForeoremThenZa+la=FonimopylZ+a1+1+n1Zneocxelpmochtiw0a)2Z(znz.)(ZralqNatuionsuesterehtsI:nemelenatgyerytaictrmeeoficnest0aa,,1...,an1Cthereexi1zts,2z,,...Cnzchsuatth(ZF=1)zkam?noisulcnocehnthegtentresnwCa
R[i],
i
More explicitly:
LetRbe the field of real numbers and letC=
2
Every complex polynomial of degreenhasncomplex roots.
1.
=
?
The fundamental theorem of algebra
Theorem
tiveffeceitekaewhtneC?foewnainalrogplyalpeapdeedl?sciohdrresis?towhehypothe
)zZ(N.)nrutaqualtiess:onthIs2z.,..z,nsCcuthhatF=(Zz1)(Zz21a,0astnna,...,reheCt11,tzisexwtiplexhcomciecoefamektifelcsuoi?nentheconstrengthC?sdewnaeredledhioworchesth?tisyhoptnehaeekwnwef?Caproolingppeaayllacirtemoegteyyarntmeleneeaer?evitcef
F=Zn+an1Zn1+∙ ∙ ∙+a1Z+a0
Theorem For every polynomial
More explicitly: LetRbe the field of real numbers and letC=R[i],i
21. =
Theorem Every complex polynomial of degreenhasncomplex roots.
The fundamental theorem of algebra
naC?ewewpgnifoorotypsiheenakehthohdrrede?sothwcinwestrenelds?Caisulcnocehtnehtgtiecffteeiak?mon
with complex coefficientsa0, a1, . . . , an1C
Theorem Every complex polynomial of degreenhasncomplex roots.
The fundamental theorem of algebra
Theorem For every polynomial F=Zn+an1Zn1+∙ ∙ ∙+a1Z+a0
More explicitly: LetRbe the field of real numbers and letC=R[i],i2=1.
ev?theeretsixz,1z..,2.,znCsuchthatF=Z(z)1Z(z)2(Na).znZuelqratuI:snoitsnaerehtsentaelemtgeoryyecilaemrteplaylpa
?
Theorem Every complex polynomial of degreenhasncomplex roots.
The fundamental theorem of algebra
itevffceieet?maksionncluheco
with complex coefficientsa0, a1, . . . , an1C there existz1, z2, . . . , znCsuch that
F= (Zz1)(Zz2)∙ ∙ ∙(Zzn).
More explicitly: LetRbe the field of real numbers and letC=R[i],i2=1.
Theorem For every polynomial F=Zn+an1Zn1+∙ ∙ ∙+a1Z+a0
is?sothwhepytoehweakenthof?CanwetgnetnehwnaCrtseeeds?ldhoicerrdoeemrtciatyreygtalingproallyappetseusnoiutaNqlarelanenemst:Irehe