the weak fundamental theorem of algebra
play

The Weak Fundamental Theorem of Algebra Robert Lubarsky Fred - PowerPoint PPT Presentation

Jan 13, 2023 •19 likes •379 views

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

  1. 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

  2. Outline The Fundamental Theorem of Algebra The Weak Fundamental Theorem of Algebra Comaximality 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

  3. Outline The Fundamental Theorem of Algebra The Weak Fundamental Theorem of Algebra Comaximality References The Fundamental Theorem of Algebra Robert Lubarsky Fred Richman Florida Atlantic University The Weak Fundamental Theorem of Algebra

  4. Outline The Fundamental Theorem of Algebra The Weak Fundamental Theorem of Algebra Comaximality References The Fundamental Theorem of Algebra Is it just true constructively? Robert Lubarsky Fred Richman Florida Atlantic University The Weak Fundamental Theorem of Algebra

  5. Outline The Fundamental Theorem of Algebra The Weak Fundamental Theorem of Algebra Comaximality 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

  6. Outline The Fundamental Theorem of Algebra The Weak Fundamental Theorem of Algebra Comaximality 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

  7. Outline The Fundamental Theorem of Algebra The Weak Fundamental Theorem of Algebra Comaximality 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

  8. Outline The Fundamental Theorem of Algebra The Weak Fundamental Theorem of Algebra Comaximality References The Fundamental Theorem of Algebra What’s the problem constructively? Robert Lubarsky Fred Richman Florida Atlantic University The Weak Fundamental Theorem of Algebra

  9. Outline The Fundamental Theorem of Algebra The Weak Fundamental Theorem of Algebra Comaximality 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

  10. Outline The Fundamental Theorem of Algebra The Weak Fundamental Theorem of Algebra Comaximality 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

  11. Outline The Fundamental Theorem of Algebra The Weak Fundamental Theorem of Algebra Comaximality 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 ′ . Robert Lubarsky Fred Richman Florida Atlantic University The Weak Fundamental Theorem of Algebra

  12. Outline The Fundamental Theorem of Algebra The Weak Fundamental Theorem of Algebra Comaximality 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

  13. Outline The Fundamental Theorem of Algebra The Weak Fundamental Theorem of Algebra Comaximality 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

  14. Outline The Fundamental Theorem of Algebra The Weak Fundamental Theorem of Algebra Comaximality References The Weak Fundamental Theorem of Algebra Theorem Let f be a nonconstant monic polynomial over C . Then the assumption that f has no roots leads to a contradiction. Robert Lubarsky Fred Richman Florida Atlantic University The Weak Fundamental Theorem of Algebra

  15. Outline The Fundamental Theorem of Algebra The Weak Fundamental Theorem of Algebra Comaximality References The Weak Fundamental Theorem of Algebra Theorem Let f be a nonconstant monic polynomial over C . Then the assumption 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

  16. Outline The Fundamental Theorem of Algebra The Weak Fundamental Theorem of Algebra Comaximality References The Weak Fundamental Theorem of Algebra Theorem Let f be a nonconstant monic polynomial over C . Then the assumption 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

  17. Outline The Fundamental Theorem of Algebra The Weak Fundamental Theorem of Algebra Comaximality References The Weak Fundamental Theorem of Algebra Theorem Let f be a nonconstant monic polynomial over C . Then the assumption 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 t with sf + tf ′ = 1. Robert Lubarsky Fred Richman Florida Atlantic University The Weak Fundamental Theorem of Algebra

  18. Outline The Fundamental Theorem of Algebra The Weak Fundamental Theorem of Algebra Comaximality References The Weak Fundamental Theorem of Algebra Theorem Let f be a nonconstant monic polynomial over C . Then the assumption 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 t with sf + tf ′ = 1. Approximate f by g = Π i ( x − q i ); note f ′ is approximated by g ′ . Robert Lubarsky Fred Richman Florida Atlantic University The Weak Fundamental Theorem of Algebra

  19. Outline The Fundamental Theorem of Algebra The Weak Fundamental Theorem of Algebra Comaximality References The Weak Fundamental Theorem of Algebra Theorem Let f be a nonconstant monic polynomial over C . Then the assumption 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 t with sf + tf ′ = 1. Approximate f by g = Π i ( x − q i ); note f ′ is approximated by g ′ . So f ( q 1 ) is close to 0, Robert Lubarsky Fred Richman Florida Atlantic University The Weak Fundamental Theorem of Algebra

  20. Outline The Fundamental Theorem of Algebra The Weak Fundamental Theorem of Algebra Comaximality References The Weak Fundamental Theorem of Algebra Theorem Let f be a nonconstant monic polynomial over C . Then the assumption 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 t with sf + tf ′ = 1. Approximate f by g = Π i ( x − q i ); note f ′ is approximated by g ′ . So f ( q 1 ) is close to 0, f ′ ( q 1 ) is bounded away from 0, Robert Lubarsky Fred Richman Florida Atlantic University The Weak Fundamental Theorem of Algebra

  21. Outline The Fundamental Theorem of Algebra The Weak Fundamental Theorem of Algebra Comaximality References The Weak Fundamental Theorem of Algebra Theorem Let f be a nonconstant monic polynomial over C . Then the assumption 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 t with sf + tf ′ = 1. Approximate f by g = Π i ( x − q i ); note f ′ is approximated by g ′ . So f ( q 1 ) is close to 0, f ′ ( q 1 ) is bounded away from 0, g ′ ( q 1 ) = Π i � =1 ( q 1 − q i ), Robert Lubarsky Fred Richman Florida Atlantic University The Weak Fundamental Theorem of Algebra

Recommend

I : Algebra : prove goal First Theorem of Algebra ) them ( Fundamental root have a . non

I : Algebra : prove goal First Theorem of Algebra ) them ( Fundamental root have a . non

Applications theorem of Fundamental I : Algebra : prove goal First Theorem of Algebra ) them ( Fundamental root have a . non constant plate EW some in All completely GI Any plx ) e Glx ) splits . p G) = O fact that , using the

227 views • 11 slides
Math 3230 Abstract Algebra I Sec 4.3: The fundamental homomorphism theorem Slides created by M.

Math 3230 Abstract Algebra I Sec 4.3: The fundamental homomorphism theorem Slides created by M.

Math 3230 Abstract Algebra I Sec 4.3: The fundamental homomorphism theorem Slides created by M. Macauley, Clemson (Modified by E. Gunawan, UConn) http://egunawan.github.io/algebra Abstract Algebra I Sec 4.3 The fundamental homomorphism theorem

506 views • 11 slides
Zeroes of polynomials and long division The Fundamental Theorem of Algebra tells us that every

Zeroes of polynomials and long division The Fundamental Theorem of Algebra tells us that every

Zeroes of polynomials and long division The Fundamental Theorem of Algebra tells us that every polynomial of Elementary Functions degree n has at most n zeroes. Part 2, Polynomials Indeed, if we are willing to count multiple zeroes and also count

768 views • 9 slides
Smales Fundamental Theorem of Algebra reconsidered Diego Armentano (joint work with Mike Shub)

Smales Fundamental Theorem of Algebra reconsidered Diego Armentano (joint work with Mike Shub)

Smales Fundamental Theorem of Algebra reconsidered Diego Armentano (joint work with Mike Shub) Centro de Matem atica, Universidad de la Rep ublica. E-mail: diego@cmat.edu.uy May 8, 2012 Fields Institute, Toronto Diego Armentano

950 views • 69 slides
The Fundamental Theorem of Algebra in ACL2 Ruben Gamboa and John Cowles Department of Computer

The Fundamental Theorem of Algebra in ACL2 Ruben Gamboa and John Cowles Department of Computer

The Fundamental Theorem of Algebra in ACL2 Ruben Gamboa and John Cowles Department of Computer Science University of Wyoming Laramie, Wyoming 82071 {ruben,cowles}@uwyo.edu ACL2 Workshop 2018 Austin, TX Outline Overview Extreme Value

657 views • 37 slides
Lecture 4.3: The fundamental homomorphism theorem Matthew Macauley Department of Mathematical

Lecture 4.3: The fundamental homomorphism theorem Matthew Macauley Department of Mathematical

Lecture 4.3: The fundamental homomorphism theorem Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4120, Modern Algebra M. Macauley (Clemson) Lecture 4.3: The fundamental

453 views • 10 slides
Lecture 6.6: The fundamental theorem of Galois theory Matthew Macauley Department of Mathematical

Lecture 6.6: The fundamental theorem of Galois theory Matthew Macauley Department of Mathematical

Lecture 6.6: The fundamental theorem of Galois theory Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4120, Modern Algebra M. Macauley (Clemson) Lecture 6.6: Fundamental

769 views • 9 slides
(Fundamental) Physics of Elementary Particles Asymmetry of weak interactions; T e GIM mechanism

(Fundamental) Physics of Elementary Particles Asymmetry of weak interactions; T e GIM mechanism

(Fundamental) Physics of Elementary Particles Asymmetry of weak interactions; T e GIM mechanism & anomaly; Weak angle; Feynman rules Tristan Hbsch Department of Physics and Astronomy Howard University, Washington DC Prirodno-Matemati

346 views • 33 slides
Math 3230 Abstract Algebra I Sec 2.4: Cayleys Theorem Slides created by M. Macauley, Clemson

Math 3230 Abstract Algebra I Sec 2.4: Cayleys Theorem Slides created by M. Macauley, Clemson

Math 3230 Abstract Algebra I Sec 2.4: Cayleys Theorem Slides created by M. Macauley, Clemson (Modified by E. Gunawan, UConn) http://egunawan.github.io/algebra Abstract Algebra I Sec 2.4 Cayleys Theorem Abstract Algebra I 1 / 8

362 views • 8 slides
II : bustling :i Galois theory ) ( Fundamental Theorem of Thur - ids.ba , The and f

II : bustling :i Galois theory ) ( Fundamental Theorem of Thur - ids.ba , The and f

Fundamental Theorem worked examples II : bustling :i Galois theory ) ( Fundamental Theorem of Thur - ids.ba , The and f satisfy A. F F functions - ::i:: Galois ift Ciii ) KIF Alk ) of is iff FIH )

111 views • 9 slides
1.1 Hellys Theorem and its Applications One of the fundamental theorems on convexity is

1.1 Hellys Theorem and its Applications One of the fundamental theorems on convexity is

1.1 Hellys Theorem and its Applications One of the fundamental theorems on convexity is Hellys Theorem , which states the following: Theorem 1 (Hellys Theorem) . Given a set C of compact convex objects in R d such that every ( d + 1) of

54 views • 4 slides
Dynamics and algebraic integers: Perspectives on Thurstons last theorem is a weak Perron

Dynamics and algebraic integers: Perspectives on Thurstons last theorem is a weak Perron

Theorem ( Thurston, Entropy in dimension one, 2014 ) There exists a post-critically finite map f : I I with entropy h(f) = log( ) Dynamics and algebraic integers: Perspectives on Thurstons last theorem is a weak Perron number. 2

426 views • 8 slides
Weak Consistency Dan Ports, CSEP 552 CAP Theorem Cant have all three of consistency,

Weak Consistency Dan Ports, CSEP 552 CAP Theorem Cant have all three of consistency,

Weak Consistency Dan Ports, CSEP 552 CAP Theorem Cant have all three of consistency, availability, and tolerance to partitions (but the devil is in the details!) CAP Eric Brewer, 2000: conjecture on reliable

418 views • 20 slides
Math 3230 Abstract Algebra I Sec 5.2: The orbit-stabilizer theorem Slides created by M. Macauley,

Math 3230 Abstract Algebra I Sec 5.2: The orbit-stabilizer theorem Slides created by M. Macauley,

Math 3230 Abstract Algebra I Sec 5.2: The orbit-stabilizer theorem Slides created by M. Macauley, Clemson (Modified by E. Gunawan, UConn) http://egunawan.github.io/algebra Abstract Algebra I Sec 5.2 The orbit-stabilizer theorem Abstract

126 views • 9 slides
MATH 12002 - CALCULUS I 4.4: Fundamental Theorem of Calculus Professor Donald L. White

MATH 12002 - CALCULUS I 4.4: Fundamental Theorem of Calculus Professor Donald L. White

MATH 12002 - CALCULUS I 4.4: Fundamental Theorem of Calculus Professor Donald L. White Department of Mathematical Sciences Kent State University D.L. White (Kent State University) 1 / 8 The Fundamental Theorem Fundamental Theorem of

290 views • 8 slides
Chapter Theorem limit Central I : tasting characterizations of weak convergence of equivalent

Chapter Theorem limit Central I : tasting characterizations of weak convergence of equivalent

Chapter Theorem limit Central I : tasting characterizations of weak convergence of equivalent Proof of distribution distribution follows theme conveyance of Observation : weak expectation interact with limits : of how IE ( " 7 Xn

522 views • 16 slides
Linear algebra version of Haj os Theorem Tommy R. Jensen Department of Mathematics Kyungpook

Linear algebra version of Haj os Theorem Tommy R. Jensen Department of Mathematics Kyungpook

Linear algebra version of Haj os Theorem Tommy R. Jensen Department of Mathematics Kyungpook National University Rep. of Korea GT2015 August 27, 2015 Tommy R. Jensen Linear algebra version of Haj os Theorem Plan for the talk

998 views • 62 slides
Definite Integrals Fundamental Theorem of Calculus Slide 3 / 85 Slide 4 / 85 Consider the

Definite Integrals Fundamental Theorem of Calculus Slide 3 / 85 Slide 4 / 85 Consider the

Slide 1 / 85 Slide 2 / 85 Table of Contents Riemann Sums Trapezoid Rule Accumulation Function Antiderivatives & Definite Integrals Mean Value Theorem & Average Value Definite Integrals Fundamental Theorem of Calculus Slide 3 / 85

402 views • 15 slides
Data Screening and Missing Value Analysis James H. Steiger Theorem (The Fundamental Theorem of

Data Screening and Missing Value Analysis James H. Steiger Theorem (The Fundamental Theorem of

Data Screening and Missing Value Analysis James H. Steiger Theorem (The Fundamental Theorem of Modern Data Analysis). Storage media are cheap. Data are expensive. Corollary 1. In some cases, the data are priceless. Corollary 2. You will suffer

579 views • 27 slides
13. The Weak Law and the Strong Law of Large Numbers James Bernoulli proved the weak law of large

13. The Weak Law and the Strong Law of Large Numbers James Bernoulli proved the weak law of large

13. The Weak Law and the Strong Law of Large Numbers James Bernoulli proved the weak law of large numbers (WLLN) around 1700 which was published posthumously in 1713 in his treatise Ars Conjectandi. Poisson generalized Bernoullis theorem

516 views • 15 slides
Amdahl s Law 18 Amdahl s Law The fundamental theorem of performance optimization

Amdahl s Law 18 Amdahl s Law The fundamental theorem of performance optimization

Amdahl s Law 18 Amdahl s Law The fundamental theorem of performance optimization Made by Amdahl! One of the designers of the IBM 360 Gave FUD it s modern meaning Optimizations do not (generally) uniformly

372 views • 19 slides
Boolean Algebra Chapter 3 Boolean Values Introduction Boolean Operations Fundamental Operators

Boolean Algebra Chapter 3 Boolean Values Introduction Boolean Operations Fundamental Operators

Boolean Algebra Chapter 3 Boolean Values Introduction Boolean Operations Fundamental Operators Boolean Operations Binary Boolean Operators Boolean Algebra Precedence of Operators Boolean Algebra Function Evaluation Boolean Algebra Basic

485 views • 32 slides
Structure theorem for a class of group - like residuated chains la Hahn Sndor Jenei

Structure theorem for a class of group - like residuated chains la Hahn Sndor Jenei

Structure theorem for a class of group - like residuated chains la Hahn Sndor Jenei University of Pcs, Hung ary FL-algebras An algebra A = ( A, , , , \ , /, 1 , 0) is called a full Lambek algebra or an FL-algebra , if ( A,

472 views • 44 slides
13 Automated Reasoning 13.0 Introduction to Weak 13.3 PROLOG and Methods in Theorem

13 Automated Reasoning 13.0 Introduction to Weak 13.3 PROLOG and Methods in Theorem

13 Automated Reasoning 13.0 Introduction to Weak 13.3 PROLOG and Methods in Theorem Automated Reasoning Proving 13.4 Further Issues in 13.1 The General Problem Automated Reasoning Solver and Difference 13.5 Epilogue and Tables

775 views • 51 slides

More recommend