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

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

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Nombre de lectures

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Cauchy Washington (District de Columbia) Polynomial Real number Álgebra Existential quantification
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:LetitlyhefieRbeterladlforeasunbm0w+ahcitplomcoexcfifetnei,0as.,1aypolynomialF=Zn+na1−nZ1−∙+∙∙a+Z11z−Z(=FtahthcusCn)−z(Z∙∙)∙z2Z−)(erxetCeh−n∈1..a,,zn∈,...1,z2istzmoegirteratnteyylieaprngllcappyauqseitnoN.tarulaeanelemes:IstherewtsC?nahtneergnordehicheldsredfihtopyhehwot?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+n−1Zn−eocxelpmochtiw0a)2∙∙Z(z−nz.)(∙−ZralqNatuionsuesterehtsI:nemelenatgyerytaictrmeeofificnest0aa,,1...,an−1∈Cthereexi1zts,2z,,...C∈nzchsuatth(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?foewnainalrogplyalpeapdeefidl?sciohdrresis?towhehypothe
∙∙∙)z−Z(N.)nrutaqualtiess:onthIs2z.,..z,∈nsCcuthhatF=(Z−z1)(Z−z21a,0astn−na,...,reheCt1∈1,tzisexwtiplexhcomficiecoefamektifelcsuoi?nentheconstrengthC?sdewnaeredlefidhioworchesth?tisyhoptnehaeekwnwef?Caproolingppeaayllacirtemoegteyyarntmeleneeaer?evitcef
F=Zn+an−1Zn−1+∙ ∙ ∙+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?sothwcinwestrenfields?Caisulcnocehtnehtgtiecffteeiak?mon
with complex coefficientsa0, a1, . . . , an−1∈C
Theorem Every complex polynomial of degreenhasncomplex roots.
The fundamental theorem of algebra
Theorem For every polynomial F=Zn+an−1Zn−1+∙ ∙ ∙+a1Z+a0
More explicitly: LetRbe the field of real numbers and letC=R[i],i2=−1.
ev?theeretsixz,1z..,2.,zn∈CsuchthatF=Z(z−)1Z(z−)2∙∙(∙Na).znZ−uelqratuI:snoitsnaerehtsentaelemtgeoryyecilaemrteplaylpa
?
Theorem Every complex polynomial of degreenhasncomplex roots.
The fundamental theorem of algebra
itevffceieet?maksionncluheco
with complex coefficientsa0, a1, . . . , an−1∈C there existz1, z2, . . . , zn∈Csuch that
F= (Z−z1)(Z−z2)∙ ∙ ∙(Z−zn).
More explicitly: LetRbe the field of real numbers and letC=R[i],i2=−1.
Theorem For every polynomial F=Zn+an−1Zn−1+∙ ∙ ∙+a1Z+a0
is?sothwhepytoehweakenthof?CanwetgnetnehwnaCrtsefieeds?ldhoicerrdoeemrtciatyreygtalingproallyappetseusnoiutaNqlarelanenemst:Irehe
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