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

Charles - Franc¸Ois Sturm

Description

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

Sujets

• Cauchy
• Washington (District de Columbia)
• Polynomial
• Real number
• Álgebra
• Existential quantification

Informations

• Publié par: pefav
• Nombre de lectures: 23
• Langue: English

Extrait

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

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More explicitly:

LetRbe the field of real numbers and letC=

2

Every complex polynomial of degreenhasncomplex roots.

−1.

=

?

The fundamental theorem of algebra

Theorem

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F=Z n +a n−1 Z n−1 +∙ ∙ ∙+a 1 Z+a 0

Theorem For every polynomial

More explicitly: LetRbe the field of real numbers and letC=R[i],i

2 1. =−

Theorem Every complex polynomial of degreenhasncomplex roots.

The fundamental theorem of algebra

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with complex coefficientsa 0 , a 1 , . . . , a n−1 ∈C

Theorem Every complex polynomial of degreenhasncomplex roots.

The fundamental theorem of algebra

Theorem For every polynomial F=Z n +a n−1 Z n−1 +∙ ∙ ∙+a 1 Z+a 0

More explicitly: LetRbe the field of real numbers and letC=R[i],i 2 =−1.

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Theorem Every complex polynomial of degreenhasncomplex roots.

The fundamental theorem of algebra

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with complex coefficientsa 0 , a 1 , . . . , a n−1 ∈C there existz 1 , z 2 , . . . , z n ∈Csuch that

F= (Z−z 1 )(Z−z 2 )∙ ∙ ∙(Z−z n ).

More explicitly: LetRbe the field of real numbers and letC=R[i],i 2 =−1.

Theorem For every polynomial F=Z n +a n−1 Z n−1 +∙ ∙ ∙+a 1 Z+a 0

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