The Weak Fundamental Theorem of Algebra
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Outline The Fundamental Theorem of Algebra The Weak Fundamental Theorem of Algebra Comaximality References The Weak Fundamental Theorem of Algebra Robert Lubarsky Fred Richman Florida Atlantic University July 27, 2009 Robert Lubarsky Fred Richman Florida Atlantic University The Weak Fundamental Theorem of Algebra
Transcript of The Weak Fundamental Theorem of Algebra
OutlineThe Fundamental Theorem of Algebra
The Weak Fundamental Theorem of AlgebraComaximality
References
The Weak Fundamental Theorem of Algebra
Robert LubarskyFred Richman
Florida Atlantic University
July 27, 2009
Robert Lubarsky Fred Richman Florida Atlantic University The Weak Fundamental Theorem of Algebra
OutlineThe Fundamental Theorem of Algebra
The Weak Fundamental Theorem of AlgebraComaximality
References
The Fundamental Theorem of Algebra
The Weak Fundamental Theorem of Algebra
Comaximality
References
Robert Lubarsky Fred Richman Florida Atlantic University The Weak Fundamental Theorem of Algebra
OutlineThe Fundamental Theorem of Algebra
The Weak Fundamental Theorem of AlgebraComaximality
References
The Fundamental Theorem of Algebra
Robert Lubarsky Fred Richman Florida Atlantic University The Weak Fundamental Theorem of Algebra
OutlineThe Fundamental Theorem of Algebra
The Weak Fundamental Theorem of AlgebraComaximality
References
The Fundamental Theorem of Algebra
Is it just true constructively?
Robert Lubarsky Fred Richman Florida Atlantic University The Weak Fundamental Theorem of Algebra
OutlineThe Fundamental Theorem of Algebra
The Weak Fundamental Theorem of AlgebraComaximality
References
The Fundamental Theorem of Algebra
Is it just true constructively?
No! Example: Sheaves over C. (Fourman-Hyland)
Robert Lubarsky Fred Richman Florida Atlantic University The Weak Fundamental Theorem of Algebra
OutlineThe Fundamental Theorem of Algebra
The Weak Fundamental Theorem of AlgebraComaximality
References
The Fundamental Theorem of Algebra
Is it just true constructively?
No! Example: Sheaves over C. (Fourman-Hyland)
Is it ever true constructively?
Robert Lubarsky Fred Richman Florida Atlantic University The Weak Fundamental Theorem of Algebra
OutlineThe Fundamental Theorem of Algebra
The Weak Fundamental Theorem of AlgebraComaximality
References
The Fundamental Theorem of Algebra
Is it just true constructively?
No! Example: Sheaves over C. (Fourman-Hyland)
Is it ever true constructively?
– Over a discrete field.
– Under Countable Choice.
– When the coefficients are Cauchy reals. (Ruitenburg)
Robert Lubarsky Fred Richman Florida Atlantic University The Weak Fundamental Theorem of Algebra
OutlineThe Fundamental Theorem of Algebra
The Weak Fundamental Theorem of AlgebraComaximality
References
The Fundamental Theorem of Algebra
What’s the problem constructively?
Robert Lubarsky Fred Richman Florida Atlantic University The Weak Fundamental Theorem of Algebra
OutlineThe Fundamental Theorem of Algebra
The Weak Fundamental Theorem of AlgebraComaximality
References
The Fundamental Theorem of Algebra
What’s the problem constructively?
– Roots that may or may not be repeated.
Robert Lubarsky Fred Richman Florida Atlantic University The Weak Fundamental Theorem of Algebra
OutlineThe Fundamental Theorem of Algebra
The Weak Fundamental Theorem of AlgebraComaximality
References
The Fundamental Theorem of Algebra
What’s the problem constructively?
– Roots that may or may not be repeated.
How can you tell when a root is repeated?
Robert Lubarsky Fred Richman Florida Atlantic University The Weak Fundamental Theorem of Algebra
OutlineThe Fundamental Theorem of Algebra
The Weak Fundamental Theorem of AlgebraComaximality
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The Fundamental Theorem of Algebra
What’s the problem constructively?
– Roots that may or may not be repeated.
How can you tell when a root is repeated?
– Compare f and its derivative f ′.
Robert Lubarsky Fred Richman Florida Atlantic University The Weak Fundamental Theorem of Algebra
OutlineThe Fundamental Theorem of Algebra
The Weak Fundamental Theorem of AlgebraComaximality
References
The Fundamental Theorem of Algebra
What’s the problem constructively?
– Roots that may or may not be repeated.
How can you tell when a root is repeated?
– Compare f and its derivative f ′.How can you see if they have a common factor?
Robert Lubarsky Fred Richman Florida Atlantic University The Weak Fundamental Theorem of Algebra
OutlineThe Fundamental Theorem of Algebra
The Weak Fundamental Theorem of AlgebraComaximality
References
The Fundamental Theorem of Algebra
What’s the problem constructively?
– Roots that may or may not be repeated.
How can you tell when a root is repeated?
– Compare f and its derivative f ′.How can you see if they have a common factor?
– The Euclidean algorithm.
Robert Lubarsky Fred Richman Florida Atlantic University The Weak Fundamental Theorem of Algebra
OutlineThe Fundamental Theorem of Algebra
The Weak Fundamental Theorem of AlgebraComaximality
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The Weak Fundamental Theorem of Algebra
TheoremLet f be a nonconstant monic polynomial over C. Then theassumption that f has no roots leads to a contradiction.
Robert Lubarsky Fred Richman Florida Atlantic University The Weak Fundamental Theorem of Algebra
OutlineThe Fundamental Theorem of Algebra
The Weak Fundamental Theorem of AlgebraComaximality
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The Weak Fundamental Theorem of Algebra
TheoremLet f be a nonconstant monic polynomial over C. Then theassumption that f has no roots leads to a contradiction.
Proof.Apply the Euclidean algorithm to f and f ′.
Robert Lubarsky Fred Richman Florida Atlantic University The Weak Fundamental Theorem of Algebra
OutlineThe Fundamental Theorem of Algebra
The Weak Fundamental Theorem of AlgebraComaximality
References
The Weak Fundamental Theorem of Algebra
TheoremLet f be a nonconstant monic polynomial over C. Then theassumption that f has no roots leads to a contradiction.
Proof.Apply the Euclidean algorithm to f and f ′. If the GCD has degree> 1, you’re done by induction.
Robert Lubarsky Fred Richman Florida Atlantic University The Weak Fundamental Theorem of Algebra
OutlineThe Fundamental Theorem of Algebra
The Weak Fundamental Theorem of AlgebraComaximality
References
The Weak Fundamental Theorem of Algebra
TheoremLet f be a nonconstant monic polynomial over C. Then theassumption that f has no roots leads to a contradiction.
Proof.Apply the Euclidean algorithm to f and f ′. If the GCD has degree> 1, you’re done by induction. Else we have polynomials s and twith sf + tf ′ = 1.
Robert Lubarsky Fred Richman Florida Atlantic University The Weak Fundamental Theorem of Algebra
OutlineThe Fundamental Theorem of Algebra
The Weak Fundamental Theorem of AlgebraComaximality
References
The Weak Fundamental Theorem of Algebra
TheoremLet f be a nonconstant monic polynomial over C. Then theassumption that f has no roots leads to a contradiction.
Proof.Apply the Euclidean algorithm to f and f ′. If the GCD has degree> 1, you’re done by induction. Else we have polynomials s and twith sf + tf ′ = 1. Approximate f by g = Πi (x − qi ); note f ′ isapproximated by g ′.
Robert Lubarsky Fred Richman Florida Atlantic University The Weak Fundamental Theorem of Algebra
OutlineThe Fundamental Theorem of Algebra
The Weak Fundamental Theorem of AlgebraComaximality
References
The Weak Fundamental Theorem of Algebra
TheoremLet f be a nonconstant monic polynomial over C. Then theassumption that f has no roots leads to a contradiction.
Proof.Apply the Euclidean algorithm to f and f ′. If the GCD has degree> 1, you’re done by induction. Else we have polynomials s and twith sf + tf ′ = 1. Approximate f by g = Πi (x − qi ); note f ′ isapproximated by g ′. So f (q1) is close to 0,
Robert Lubarsky Fred Richman Florida Atlantic University The Weak Fundamental Theorem of Algebra
OutlineThe Fundamental Theorem of Algebra
The Weak Fundamental Theorem of AlgebraComaximality
References
The Weak Fundamental Theorem of Algebra
TheoremLet f be a nonconstant monic polynomial over C. Then theassumption that f has no roots leads to a contradiction.
Proof.Apply the Euclidean algorithm to f and f ′. If the GCD has degree> 1, you’re done by induction. Else we have polynomials s and twith sf + tf ′ = 1. Approximate f by g = Πi (x − qi ); note f ′ isapproximated by g ′. So f (q1) is close to 0, f ′(q1) is bounded awayfrom 0,
Robert Lubarsky Fred Richman Florida Atlantic University The Weak Fundamental Theorem of Algebra
OutlineThe Fundamental Theorem of Algebra
The Weak Fundamental Theorem of AlgebraComaximality
References
The Weak Fundamental Theorem of Algebra
TheoremLet f be a nonconstant monic polynomial over C. Then theassumption that f has no roots leads to a contradiction.
Proof.Apply the Euclidean algorithm to f and f ′. If the GCD has degree> 1, you’re done by induction. Else we have polynomials s and twith sf + tf ′ = 1. Approximate f by g = Πi (x − qi ); note f ′ isapproximated by g ′. So f (q1) is close to 0, f ′(q1) is bounded awayfrom 0, g ′(q1) = Πi �=1(q1 − qi),
Robert Lubarsky Fred Richman Florida Atlantic University The Weak Fundamental Theorem of Algebra
OutlineThe Fundamental Theorem of Algebra
The Weak Fundamental Theorem of AlgebraComaximality
References
The Weak Fundamental Theorem of Algebra
TheoremLet f be a nonconstant monic polynomial over C. Then theassumption that f has no roots leads to a contradiction.
Proof.Apply the Euclidean algorithm to f and f ′. If the GCD has degree> 1, you’re done by induction. Else we have polynomials s and twith sf + tf ′ = 1. Approximate f by g = Πi (x − qi ); note f ′ isapproximated by g ′. So f (q1) is close to 0, f ′(q1) is bounded awayfrom 0, g ′(q1) = Πi �=1(q1 − qi), and q1 is bounded away from eachof the other qi ’s.
Robert Lubarsky Fred Richman Florida Atlantic University The Weak Fundamental Theorem of Algebra
OutlineThe Fundamental Theorem of Algebra
The Weak Fundamental Theorem of AlgebraComaximality
References
The Weak Fundamental Theorem of Algebra
TheoremLet f be a nonconstant monic polynomial over C. Then theassumption that f has no roots leads to a contradiction.
Proof.Apply the Euclidean algorithm to f and f ′. If the GCD has degree> 1, you’re done by induction. Else we have polynomials s and twith sf + tf ′ = 1. Approximate f by g = Πi (x − qi ); note f ′ isapproximated by g ′. So f (q1) is close to 0, f ′(q1) is bounded awayfrom 0, g ′(q1) = Πi �=1(q1 − qi), and q1 is bounded away from eachof the other qi ’s. So we have coherent approximations to the rootnear q1.
Robert Lubarsky Fred Richman Florida Atlantic University The Weak Fundamental Theorem of Algebra
OutlineThe Fundamental Theorem of Algebra
The Weak Fundamental Theorem of AlgebraComaximality
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The Weak Fundamental Theorem of Algebra
So the root set S of a polynomial may not be inhabited, but itcan’t be empty.
Robert Lubarsky Fred Richman Florida Atlantic University The Weak Fundamental Theorem of Algebra
OutlineThe Fundamental Theorem of Algebra
The Weak Fundamental Theorem of AlgebraComaximality
References
The Weak Fundamental Theorem of Algebra
So the root set S of a polynomial may not be inhabited, but itcan’t be empty. In fact,
� the distance d(z ,S) = infx∈S d(z , x) may not be defined, butthe quasi-distance δ(z ,S) = supx∈S d(z , x) is;
Robert Lubarsky Fred Richman Florida Atlantic University The Weak Fundamental Theorem of Algebra
OutlineThe Fundamental Theorem of Algebra
The Weak Fundamental Theorem of AlgebraComaximality
References
The Weak Fundamental Theorem of Algebra
So the root set S of a polynomial may not be inhabited, but itcan’t be empty. In fact,
� the distance d(z ,S) = infx∈S d(z , x) may not be defined, butthe quasi-distance δ(z ,S) = supx∈S d(z , x) is;
� S may not be finite, but it’s quasi-finite; and
Robert Lubarsky Fred Richman Florida Atlantic University The Weak Fundamental Theorem of Algebra
OutlineThe Fundamental Theorem of Algebra
The Weak Fundamental Theorem of AlgebraComaximality
References
The Weak Fundamental Theorem of Algebra
So the root set S of a polynomial may not be inhabited, but itcan’t be empty. In fact,
� the distance d(z ,S) = infx∈S d(z , x) may not be defined, butthe quasi-distance δ(z ,S) = supx∈S d(z , x) is;
� S may not be finite, but it’s quasi-finite; and
� there is a Riesz space of functions on S .
Robert Lubarsky Fred Richman Florida Atlantic University The Weak Fundamental Theorem of Algebra
OutlineThe Fundamental Theorem of Algebra
The Weak Fundamental Theorem of AlgebraComaximality
References
Comaximality
DefinitionIn a ring R , a and b are comaximal if the ideal (a, b) equals R ; i.e.for some s, t ∈ R sa + tb = 1.
Robert Lubarsky Fred Richman Florida Atlantic University The Weak Fundamental Theorem of Algebra
OutlineThe Fundamental Theorem of Algebra
The Weak Fundamental Theorem of AlgebraComaximality
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Comaximality
DefinitionIn a ring R , a and b are comaximal if the ideal (a, b) equals R ; i.e.for some s, t ∈ R sa + tb = 1.
DefinitionThe resultant of polynomials a(x) and b(x), Res(a, b), is thedeterminant of the Sylvester matrix.
Robert Lubarsky Fred Richman Florida Atlantic University The Weak Fundamental Theorem of Algebra
OutlineThe Fundamental Theorem of Algebra
The Weak Fundamental Theorem of AlgebraComaximality
References
Comaximality
DefinitionIn a ring R , a and b are comaximal if the ideal (a, b) equals R ; i.e.for some s, t ∈ R sa + tb = 1.
DefinitionThe resultant of polynomials a(x) and b(x), Res(a, b), is thedeterminant of the Sylvester matrix.
Example: For a = Σiaixi and b = Σjbjx
j , the Sylvester matrix is
a2 a1 a0 0 0 00 a2 a1 a0 0 00 0 a2 a1 a0 00 0 0 a2 a1 a0
b4 b3 b2 b1 b0 00 b4 b3 b2 b1 b0
.
Robert Lubarsky Fred Richman Florida Atlantic University The Weak Fundamental Theorem of Algebra
OutlineThe Fundamental Theorem of Algebra
The Weak Fundamental Theorem of AlgebraComaximality
References
Comaximality
DefinitionIn a ring R , a and b are comaximal if the ideal (a, b) equals R ; i.e.for some s, t ∈ R sa + tb = 1.
DefinitionThe resultant of polynomials a(x) and b(x), Res(a, b), is thedeterminant of the Sylvester matrix.
Standard Facts If a = Π(x − qi ) and b = Π(x − rj ), then Res(a, b)= Π(qi − rj).
Robert Lubarsky Fred Richman Florida Atlantic University The Weak Fundamental Theorem of Algebra
OutlineThe Fundamental Theorem of Algebra
The Weak Fundamental Theorem of AlgebraComaximality
References
Comaximality
DefinitionIn a ring R , a and b are comaximal if the ideal (a, b) equals R ; i.e.for some s, t ∈ R sa + tb = 1.
DefinitionThe resultant of polynomials a(x) and b(x), Res(a, b), is thedeterminant of the Sylvester matrix.
Standard Facts If a = Π(x − qi ) and b = Π(x − rj ), then Res(a, b)= Π(qi − rj). So over a discrete field, Res(a, b) = 0 iff a and bhave a common non-trivial factor.
Robert Lubarsky Fred Richman Florida Atlantic University The Weak Fundamental Theorem of Algebra
OutlineThe Fundamental Theorem of Algebra
The Weak Fundamental Theorem of AlgebraComaximality
References
Comaximality
DefinitionIn a ring R , a and b are comaximal if the ideal (a, b) equals R ; i.e.for some s, t ∈ R sa + tb = 1.
DefinitionThe resultant of polynomials a(x) and b(x), Res(a, b), is thedeterminant of the Sylvester matrix.
Standard Facts If a = Π(x − qi ) and b = Π(x − rj ), then Res(a, b)= Π(qi − rj). So over a discrete field, Res(a, b) = 0 iff a and bhave a common non-trivial factor.This doesn’t generalize well to arbitrary rings. Example: Over Z8,x2 + 4 and x2 + 4x have a resultant of 0 but no non-trivialcommon factors. Hence:
Robert Lubarsky Fred Richman Florida Atlantic University The Weak Fundamental Theorem of Algebra
OutlineThe Fundamental Theorem of Algebra
The Weak Fundamental Theorem of AlgebraComaximality
References
Comaximality
DefinitionIn a ring R , a and b are comaximal if the ideal (a, b) equals R ; i.e.for some s, t ∈ R sa + tb = 1.
DefinitionThe resultant of polynomials a(x) and b(x), Res(a, b), is thedeterminant of the Sylvester matrix.
Standard Facts If a = Π(x − qi ) and b = Π(x − rj ), then Res(a, b)= Π(qi − rj). So over a discrete field, Res(a, b) = 0 iff a and bhave a common non-trivial factor.
TheoremFor a and b monic, Res(a, b) is a unit iff a and b are comaximal.
Robert Lubarsky Fred Richman Florida Atlantic University The Weak Fundamental Theorem of Algebra
OutlineThe Fundamental Theorem of Algebra
The Weak Fundamental Theorem of AlgebraComaximality
References
Comaximality
DefinitionIn a ring R , a and b are comaximal if the ideal (a, b) equals R ; i.e.for some s, t ∈ R sa + tb = 1.
DefinitionThe resultant of polynomials a(x) and b(x), Res(a, b), is thedeterminant of the Sylvester matrix.
TheoremFor a and b monic, Res(a, b) is a unit iff a and b are comaximal.
Corollary
For a and b monic polynomials over C, Res(a, b) is a unit iff a andb are comaximal iff there is a positive distance between the rootsof a and b.
Robert Lubarsky Fred Richman Florida Atlantic University The Weak Fundamental Theorem of Algebra
OutlineThe Fundamental Theorem of Algebra
The Weak Fundamental Theorem of AlgebraComaximality
References
References
� Fourman/Hyland, Sheaf models for analysis, in M.P. Fourman,C.J. Mulvey, and D.S. Scott (eds.), Applications of Sheaves,Lecture Notes in Mathematics Vol. 753, Springer-Verlag,Berlin Heidelberg New York, 1979, p. 280-301
� Lubarsky/Richman, Zero Sets of Univariate Polynomials,TAMS, to appear
� Richman, The Fundamental Theorem of Algebra: aConstructive Development without Choice, Pacific Journal ofMathematics, v. 196, 2000, p. 213-230
� Ruitenburg, Constructing roots of polynomials over thecomplex numbers, Computational aspects of Lie grouprepresentations and related topics (Amsterdam 1996)107–128, CWI Tract, 84, Math. Centrum, Centrum Wisk.Inform., Amsterdam 1991. MR 92g:03085
Robert Lubarsky Fred Richman Florida Atlantic University The Weak Fundamental Theorem of Algebra
