solving polynomials after klein the theory of resolvent
play

Solving Polynomials After Klein: The Theory of Resolvent Degree - PowerPoint PPT Presentation

Aug 25, 2022 •10 likes •1.4k views

Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Kleins Solution to the Quintic The Sextic Alexander J. Sutherland University of California, Irvine Department of Mathematics Solving

  1. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Quadratic Formula - Topology Figure 1: A 2 -sheeted branched cover of S 1 Image from Christoper Dustin’s blog, ”Representing Spacetime as a Branched Covering Space”, (Link) May 16 th , 2019 Alex Sutherland 15 / 74

  2. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Quadratic Formula - Topology C 1 E 1 z �� z 2 C 2 C 1 ( b,c ) b 2 − 4 c Why is E 1 � C 2 ( b,c ) a branched cover? When b 2 − 4 c = 0 , the fiber collapses to a point ( z 2 + bz + c has a unique root with multiplicity 2) May 16 th , 2019 Alex Sutherland 16 / 74

  3. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Quadratic Formula - Topology C 1 E 1 z �� z 2 C 2 C 1 ( b,c ) b 2 − 4 c Why is E 1 � C 2 ( b,c ) a branched cover? When b 2 − 4 c = 0 , the fiber collapses to a point ( z 2 + bz + c has a unique root with multiplicity 2) May 16 th , 2019 Alex Sutherland 16 / 74

  4. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Complex Varieties Definition A complex variety ( variety over C ) is a reduced scheme of finite type over Spec ( C ) . Varieties are reduced, but may not be irreducible. May 16 th , 2019 Alex Sutherland 17 / 74

  5. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Categories We define two categories: • IrrVars/ C - objects are irreducible complex varieties, morphisms are dominant rational maps • Fields/ C - objects are field extensions of C with finite transcendence degree, morphisms are field embeddings May 16 th , 2019 Alex Sutherland 18 / 74

  6. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Equivalences of Categories Lemma The functor induced by C : IrrVars/ C op � Fields/ C X �� C ( X ) is an equivalence of categories. May 16 th , 2019 Alex Sutherland 19 / 74

  7. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Equivalences of Categories Corollary The induced functor on arrow categories Ar ( C ) : Ar ( IrrVars/ C op ) � Ar ( Fields/ C ) ( Y � X ) �� ( C ( X ) ֒ � C ( Y )) is an equivalence of categories. Takeaway: Today - branched covers of complex varieties. Can also tell the same story in terms of field extensions May 16 th , 2019 Alex Sutherland 20 / 74

  8. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Equivalences of Categories Corollary The induced functor on arrow categories Ar ( C ) : Ar ( IrrVars/ C op ) � Ar ( Fields/ C ) ( Y � X ) �� ( C ( X ) ֒ � C ( Y )) is an equivalence of categories. Takeaway: Today - branched covers of complex varieties. Can also tell the same story in terms of field extensions May 16 th , 2019 Alex Sutherland 20 / 74

  9. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Algebraic Functions Definition Let X be a complex variety. An algebraic function on X is an n -valued function ϕ : X � C { } z | z n + a 1 ( x ) z n − 1 + · · · + a n ( x ) = 0 x �� where each a i is a rational function on X May 16 th , 2019 Alex Sutherland 21 / 74

  10. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Algebraic Functions Example Let X be the complex variety C n and define a i to be the i th coordinate function a i : X � C x �� x i Define the algebraic function Φ n as follows: Φ n : X � C { } z | z n + a 1 ( x ) z n − 1 + · · · + a n ( x ) = 0 x �� May 16 th , 2019 Alex Sutherland 22 / 74

  11. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Algebraic Functions Example Let X be the complex variety C n and define a i to be the i th coordinate function a i : X � C x �� x i Define the algebraic function Φ n as follows: Φ n : X � C { } z | z n + a 1 ( x ) z n − 1 + · · · + a n ( x ) = 0 x �� May 16 th , 2019 Alex Sutherland 22 / 74

  12. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Restating our Classical Question Re-state classical question in this language: Classical Question (Re-stated) Give a formula for Φ n . What is a formula for an algebraic function? Generalization of the topological version of quadratic formula May 16 th , 2019 Alex Sutherland 23 / 74

  13. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Restating our Classical Question Re-state classical question in this language: Classical Question (Re-stated) Give a formula for Φ n . What is a formula for an algebraic function? Generalization of the topological version of quadratic formula May 16 th , 2019 Alex Sutherland 23 / 74

  14. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Restating our Classical Question Re-state classical question in this language: Classical Question (Re-stated) Give a formula for Φ n . What is a formula for an algebraic function? Generalization of the topological version of quadratic formula May 16 th , 2019 Alex Sutherland 23 / 74

  15. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Formula for a Branched Cover Given a branched cover of complex varieties Y � X , a formula in functions of d variables for Y � X of length r is a finite tower of branched covers of complex varieties X r � X r − 1 � · · · � X 1 � X 0 ⊆ X such that • X 0 ⊆ X is a dense Zariski open, • X r � X factors through a branched cover X r � Y , • each map X i � X i − 1 comes from a pullback square of complex varieties of dimension at most d . May 16 th , 2019 Alex Sutherland 24 / 74

  16. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Formula for a Branched Cover Given a branched cover of complex varieties Y � X , a formula in functions of d variables for Y � X of length r is a finite tower of branched covers of complex varieties X r � X r − 1 � · · · � X 1 � X 0 ⊆ X such that • X 0 ⊆ X is a dense Zariski open, • X r � X factors through a branched cover X r � Y , • each map X i � X i − 1 comes from a pullback square of complex varieties of dimension at most d . May 16 th , 2019 Alex Sutherland 24 / 74

  17. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Formula for a Branched Cover Given a branched cover of complex varieties Y � X , a formula in functions of d variables for Y � X of length r is a finite tower of branched covers of complex varieties X r � X r − 1 � · · · � X 1 � X 0 ⊆ X such that • X 0 ⊆ X is a dense Zariski open, • X r � X factors through a branched cover X r � Y , • each map X i � X i − 1 comes from a pullback square of complex varieties of dimension at most d . May 16 th , 2019 Alex Sutherland 24 / 74

  18. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Formula for a Branched Cover Given a branched cover of complex varieties Y � X , a formula in functions of d variables for Y � X of length r is a finite tower of branched covers of complex varieties X r � X r − 1 � · · · � X 1 � X 0 ⊆ X such that • X 0 ⊆ X is a dense Zariski open, • X r � X factors through a branched cover X r � Y , • each map X i � X i − 1 comes from a pullback square of complex varieties of dimension at most d . May 16 th , 2019 Alex Sutherland 24 / 74

  19. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Formula for a Branched Cover Given a branched cover of complex varieties Y � X , a formula in functions of d variables for Y � X of length r is a finite tower of branched covers of complex varieties X r � X r − 1 � · · · � X 1 � X 0 ⊆ X such that • X 0 ⊆ X is a dense Zariski open, • X r � X factors through a branched cover X r � Y , • each map X i � X i − 1 comes from a pullback square of complex varieties of dimension at most d . May 16 th , 2019 Alex Sutherland 24 / 74

  20. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Formula for a Branched Cover Given a branched cover of complex varieties Y � X , a formula in functions of d variables for Y � X of length r is a finite tower of branched covers of complex varieties X r � X r − 1 � · · · � X 1 � X 0 ⊆ X such that • X 0 ⊆ X is a dense Zariski open, • X r � X factors through a branched cover X r � Y , • each map X i � X i − 1 comes from a pullback square of complex varieties of dimension at most d . May 16 th , 2019 Alex Sutherland 24 / 74

  21. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Formula for a Branched Cover Given a branched cover of complex varieties Y � X , a formula in functions of d variables for Y � X of length r is a finite tower of branched covers of complex varieties X r � X r − 1 � · · · � X 1 � X 0 ⊆ X such that • X 0 ⊆ X is a dense Zariski open, • X r � X factors through a branched cover X r � Y , • each map X i � X i − 1 comes from a pullback square of complex varieties of dimension at most d . May 16 th , 2019 Alex Sutherland 24 / 74

  22. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Formula for an Algebraic Function How does this help us define formulas for algebraic functions? Given an algebraic function ϕ , we construct a canonical branched cover May 16 th , 2019 Alex Sutherland 25 / 74

  23. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Construction of a Branched Cover Associated to an Algebraic Function Let X be a complex variety and ϕ an algebraic function on X given by { } z | z n + a 1 ( x ) z n − 1 + · · · + a n ( x ) = 0 x �� . Explicitly write a i ( x ) = f i ( x ) g i ( x ) and set U = X \ Z ( g 1 , . . . , g n ) . Construct E φ = { ( x, z ) ∈ U × P 1 | z n + a 1 ( x ) z n − 1 + · · · + a n ( x ) = 0 } ⊆ X × P 1 . Get branched cover E φ � X given by ( x, z ) �� x . May 16 th , 2019 Alex Sutherland 26 / 74

  24. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Construction of a Branched Cover Associated to an Algebraic Function Let X be a complex variety and ϕ an algebraic function on X given by { } z | z n + a 1 ( x ) z n − 1 + · · · + a n ( x ) = 0 x �� . Explicitly write a i ( x ) = f i ( x ) g i ( x ) and set U = X \ Z ( g 1 , . . . , g n ) . Construct E φ = { ( x, z ) ∈ U × P 1 | z n + a 1 ( x ) z n − 1 + · · · + a n ( x ) = 0 } ⊆ X × P 1 . Get branched cover E φ � X given by ( x, z ) �� x . May 16 th , 2019 Alex Sutherland 26 / 74

  25. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Construction of a Branched Cover Associated to an Algebraic Function Let X be a complex variety and ϕ an algebraic function on X given by { } z | z n + a 1 ( x ) z n − 1 + · · · + a n ( x ) = 0 x �� . Explicitly write a i ( x ) = f i ( x ) g i ( x ) and set U = X \ Z ( g 1 , . . . , g n ) . Construct E φ = { ( x, z ) ∈ U × P 1 | z n + a 1 ( x ) z n − 1 + · · · + a n ( x ) = 0 } ⊆ X × P 1 . Get branched cover E φ � X given by ( x, z ) �� x . May 16 th , 2019 Alex Sutherland 26 / 74

  26. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Construction of a Branched Cover Associated to an Algebraic Function Let X be a complex variety and ϕ an algebraic function on X given by { } z | z n + a 1 ( x ) z n − 1 + · · · + a n ( x ) = 0 x �� . Explicitly write a i ( x ) = f i ( x ) g i ( x ) and set U = X \ Z ( g 1 , . . . , g n ) . Construct E φ = { ( x, z ) ∈ U × P 1 | z n + a 1 ( x ) z n − 1 + · · · + a n ( x ) = 0 } ⊆ X × P 1 . Get branched cover E φ � X given by ( x, z ) �� x . May 16 th , 2019 Alex Sutherland 26 / 74

  27. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Formula for an Algebraic Function Definition Let ϕ be an algebraic function on a complex variety X . A formula for ϕ is a formula for the branched cover E φ � X. Want a formula for Φ n . Moreover, want the formula to be as simple as possible. Need to make this precise. May 16 th , 2019 Alex Sutherland 27 / 74

  28. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Formula for an Algebraic Function Definition Let ϕ be an algebraic function on a complex variety X . A formula for ϕ is a formula for the branched cover E φ � X. Want a formula for Φ n . Moreover, want the formula to be as simple as possible. Need to make this precise. May 16 th , 2019 Alex Sutherland 27 / 74

  29. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Formula for an Algebraic Function Definition Let ϕ be an algebraic function on a complex variety X . A formula for ϕ is a formula for the branched cover E φ � X. Want a formula for Φ n . Moreover, want the formula to be as simple as possible. Need to make this precise. May 16 th , 2019 Alex Sutherland 27 / 74

  30. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Formula for an Algebraic Function Definition Let ϕ be an algebraic function on a complex variety X . A formula for ϕ is a formula for the branched cover E φ � X. Want a formula for Φ n . Moreover, want the formula to be as simple as possible. Need to make this precise. May 16 th , 2019 Alex Sutherland 27 / 74

  31. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Resolvent Degree and Essential Dimension Definition An n -sheeted cover Y � X is defined over a variety X 0 if there is an n -sheeted cover Y 0 � X 0 such that Y ∼ = Y 0 × X 0 X for some map X � X 0 . The essential dimension of Y � X is ed ( Y � X ) = min { dim( X 0 ) | Y � X is defined over X 0 } . May 16 th , 2019 Alex Sutherland 28 / 74

  32. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Resolvent Degree and Essential Dimension Equivalently, the essential dimension of Y � X is ed ( Y � X ) = min { d | ∃ a formula of length 1 in d variables } . . May 16 th , 2019 Alex Sutherland 29 / 74

  33. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Resolvent Degree and Essential Dimension Definition The resolvent degree of Y � X is RD ( Y � X ) = min { d | ∃ a formula in d variables } May 16 th , 2019 Alex Sutherland 30 / 74

  34. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Resolvent Degree and Essential Dimension Given an algebraic function ϕ on X , the essential dimension / resolvent degree of ϕ is the essential dimension / resolvent degree of E φ � X . • ed ( ϕ ) - how simply we can write ϕ • RD ( ϕ ) - how simply we can write a formula for ϕ ed ( n ) := ed (Φ n ) RD ( n ) := RD (Φ n ) May 16 th , 2019 Alex Sutherland 31 / 74

  35. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Resolvent Degree and Essential Dimension Given an algebraic function ϕ on X , the essential dimension / resolvent degree of ϕ is the essential dimension / resolvent degree of E φ � X . • ed ( ϕ ) - how simply we can write ϕ • RD ( ϕ ) - how simply we can write a formula for ϕ ed ( n ) := ed (Φ n ) RD ( n ) := RD (Φ n ) May 16 th , 2019 Alex Sutherland 31 / 74

  36. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Resolvent Degree and Essential Dimension Given an algebraic function ϕ on X , the essential dimension / resolvent degree of ϕ is the essential dimension / resolvent degree of E φ � X . • ed ( ϕ ) - how simply we can write ϕ • RD ( ϕ ) - how simply we can write a formula for ϕ ed ( n ) := ed (Φ n ) RD ( n ) := RD (Φ n ) May 16 th , 2019 Alex Sutherland 31 / 74

  37. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Examples of RD and ed What do we know? 1 2 3 4 5 n ed ( n ) RD ( n ) May 16 th , 2019 Alex Sutherland 32 / 74

  38. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Examples of RD and ed What do we know? 1 2 3 4 5 n ed ( n ) 1 RD ( n ) 1 May 16 th , 2019 Alex Sutherland 33 / 74

  39. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Examples of RD and ed What do we know? 1 2 3 4 5 n ed ( n ) 1 1 RD ( n ) 1 1 May 16 th , 2019 Alex Sutherland 34 / 74

  40. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Examples of RD and ed What do we know? 1 2 3 4 5 n ed ( n ) 1 1 1 2 RD ( n ) 1 1 1 1 May 16 th , 2019 Alex Sutherland 35 / 74

  41. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Examples of RD and ed What do we know? 1 2 3 4 5 n ed ( n ) 1 1 1 2 2 RD ( n ) 1 1 1 1 1 ”Kronecker’s Theorem” - Felix Klein Solving quintic in one step requires functions of two variables Using longer towers, only need functions of one variable May 16 th , 2019 Alex Sutherland 36 / 74

  42. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Examples of RD and ed What do we know? 1 2 3 4 5 n ed ( n ) 1 1 1 2 2 RD ( n ) 1 1 1 1 1 ”Kronecker’s Theorem” - Felix Klein Solving quintic in one step requires functions of two variables Using longer towers, only need functions of one variable May 16 th , 2019 Alex Sutherland 36 / 74

  43. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Examples of RD and ed What do we know? 1 2 3 4 5 n ed ( n ) 1 1 1 2 2 RD ( n ) 1 1 1 1 1 ”Kronecker’s Theorem” - Felix Klein Solving quintic in one step requires functions of two variables Using longer towers, only need functions of one variable May 16 th , 2019 Alex Sutherland 36 / 74

  44. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Upper Bounds on RD Essential dimension ̸ = resolvent degree Focus on resolvent degree Upper bounds on resolvent degree: RD (5) = 1 (Bring, Klein) 1 2 3 4 5 6 7 8 9 n RD ( n ) 1 1 1 1 1 May 16 th , 2019 Alex Sutherland 37 / 74

  45. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Upper Bounds on RD Essential dimension ̸ = resolvent degree Focus on resolvent degree Upper bounds on resolvent degree: RD (5) = 1 (Bring, Klein) 1 2 3 4 5 6 7 8 9 n RD ( n ) 1 1 1 1 1 May 16 th , 2019 Alex Sutherland 37 / 74

  46. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Upper Bounds on RD Essential dimension ̸ = resolvent degree Focus on resolvent degree Upper bounds on resolvent degree: RD (5) = 1 (Bring, Klein) 1 2 3 4 5 6 7 8 9 n RD ( n ) 1 1 1 1 1 May 16 th , 2019 Alex Sutherland 37 / 74

  47. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Upper Bounds on RD Essential dimension ̸ = resolvent degree Focus on resolvent degree Upper bounds on resolvent degree: RD (5) = 1 (Bring, Klein) 1 2 3 4 5 6 7 8 9 n RD ( n ) 1 1 1 1 1 May 16 th , 2019 Alex Sutherland 37 / 74

  48. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Upper Bounds on RD Essential dimension ̸ = resolvent degree Focus on resolvent degree Upper bounds on resolvent degree: RD (6) ≤ 2 (Hamilton, Klein) 1 2 3 4 5 6 7 8 9 n RD ( n ) 1 1 1 1 1 ≤ 2 May 16 th , 2019 Alex Sutherland 38 / 74

  49. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Upper Bounds on RD Essential dimension ̸ = resolvent degree Focus on resolvent degree Upper bounds on resolvent degree: RD (7) ≤ 3 (Hamilton, Klein) 1 2 3 4 5 6 7 8 9 n RD ( n ) 1 1 1 1 1 ≤ 2 ≤ 3 May 16 th , 2019 Alex Sutherland 39 / 74

  50. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Upper Bounds on RD Essential dimension ̸ = resolvent degree Focus on resolvent degree Upper bounds on resolvent degree: RD (8) ≤ 4 (Hamilton) 1 2 3 4 5 6 7 8 9 n RD ( n ) 1 1 1 1 1 ≤ 2 ≤ 3 ≤ 4 May 16 th , 2019 Alex Sutherland 40 / 74

  51. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Upper Bounds on RD Essential dimension ̸ = resolvent degree Focus on resolvent degree Upper bounds on resolvent degree: RD (9) ≤ 4 (Hilbert) 1 2 3 4 5 6 7 8 9 n RD ( n ) 1 1 1 1 1 ≤ 2 ≤ 3 ≤ 4 ≤ 4 May 16 th , 2019 Alex Sutherland 41 / 74

  52. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Hilbert’s Conjectures Translate Hilbert’s conjectures into modern language: • Hilbert’s Sextic Conjecture: RD (6) = 2 • Hilbert’s 13th Problem: RD (7) = 3 • Hilbert’s Octic Conjecture: RD (8) = 4 1 2 3 4 5 6 7 8 9 n RD ( n ) 1 1 1 1 1 ≤ 2 ≤ 3 ≤ 4 ≤ 4 May 16 th , 2019 Alex Sutherland 42 / 74

  53. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Hilbert’s Conjectures Translate Hilbert’s conjectures into modern language: • Hilbert’s Sextic Conjecture: RD (6) = 2 • Hilbert’s 13th Problem: RD (7) = 3 • Hilbert’s Octic Conjecture: RD (8) = 4 1 2 3 4 5 6 7 8 9 n RD ( n ) 1 1 1 1 1 ≤ 2 ≤ 3 ≤ 4 ≤ 4 May 16 th , 2019 Alex Sutherland 42 / 74

  54. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Hilbert’s Conjectures Translate Hilbert’s conjectures into modern language: • Hilbert’s Sextic Conjecture: RD (6) = 2 • Hilbert’s 13th Problem: RD (7) = 3 • Hilbert’s Octic Conjecture: RD (8) = 4 1 2 3 4 5 6 7 8 9 n RD ( n ) 1 1 1 1 1 ≤ 2 ≤ 3 ≤ 4 ≤ 4 May 16 th , 2019 Alex Sutherland 42 / 74

  55. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Hilbert’s Conjectures Translate Hilbert’s conjectures into modern language: • Hilbert’s Sextic Conjecture: RD (6) = 2 • Hilbert’s 13th Problem: RD (7) = 3 • Hilbert’s Octic Conjecture: RD (8) = 4 1 2 3 4 5 6 7 8 9 n RD ( n ) 1 1 1 1 1 ≤ 2 ≤ 3 ≤ 4 ≤ 4 May 16 th , 2019 Alex Sutherland 42 / 74

  56. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Bounds on Resolvent Degree Hamilton (1836), Sylvester (1887), Brauer (1975), and Wolfson - Upper bounds on RD ( n ) Bounds not expected to be sharp (for large n ) No (non-trivial) lower bounds on RD ( n ) In particular, unknown if RD ( n ) ≡ 1 However, expect RD ( n ) � ∞ as n � ∞ May 16 th , 2019 Alex Sutherland 43 / 74

  57. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Bounds on Resolvent Degree Hamilton (1836), Sylvester (1887), Brauer (1975), and Wolfson - Upper bounds on RD ( n ) Bounds not expected to be sharp (for large n ) No (non-trivial) lower bounds on RD ( n ) In particular, unknown if RD ( n ) ≡ 1 However, expect RD ( n ) � ∞ as n � ∞ May 16 th , 2019 Alex Sutherland 43 / 74

  58. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Bounds on Resolvent Degree Hamilton (1836), Sylvester (1887), Brauer (1975), and Wolfson - Upper bounds on RD ( n ) Bounds not expected to be sharp (for large n ) No (non-trivial) lower bounds on RD ( n ) In particular, unknown if RD ( n ) ≡ 1 However, expect RD ( n ) � ∞ as n � ∞ May 16 th , 2019 Alex Sutherland 43 / 74

  59. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Bounds on Resolvent Degree Hamilton (1836), Sylvester (1887), Brauer (1975), and Wolfson - Upper bounds on RD ( n ) Bounds not expected to be sharp (for large n ) No (non-trivial) lower bounds on RD ( n ) In particular, unknown if RD ( n ) ≡ 1 However, expect RD ( n ) � ∞ as n � ∞ May 16 th , 2019 Alex Sutherland 43 / 74

  60. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Bounds on Resolvent Degree Hamilton (1836), Sylvester (1887), Brauer (1975), and Wolfson - Upper bounds on RD ( n ) Bounds not expected to be sharp (for large n ) No (non-trivial) lower bounds on RD ( n ) In particular, unknown if RD ( n ) ≡ 1 However, expect RD ( n ) � ∞ as n � ∞ May 16 th , 2019 Alex Sutherland 43 / 74

  61. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Quote by Dixmier Conclusion to Dixmier’s summary on Hilbert’s 13th problem 1 ”Let’s end on a dramatic note, which proves our incredible ignorance. Although this seems unlikely, it is not impossible that RD ( n ) = 1 for all n ! . . . Any reduction of RD ( n ) would be serious progress. In particular, it is time to know whether RD (6) = 1 or RD (6) = 2 .” 1 J. Dixmier, ”Histoire du 13 e problème de Hilbert,” in: Analyse diophantienne et géom’etrie algébrique, Cahiers Sém. Hist. Math., Sér 2, vol. 3, Univ. Paris VI, Paris, 1993, p85-94. May 16 th , 2019 Alex Sutherland 44 / 74

  62. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Remaining Goals for the Talk Formulas for Sextic - Hamilton, detailed sketch by Klein Research Goal - Understand precise geometric relationship between solutions of Hamilton and Klein Hopefully gives insight to resolvent degree, as well. Start with Klein’s solution of the quintic. May 16 th , 2019 Alex Sutherland 45 / 74

  63. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Remaining Goals for the Talk Formulas for Sextic - Hamilton, detailed sketch by Klein Research Goal - Understand precise geometric relationship between solutions of Hamilton and Klein Hopefully gives insight to resolvent degree, as well. Start with Klein’s solution of the quintic. May 16 th , 2019 Alex Sutherland 45 / 74

  64. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Remaining Goals for the Talk Formulas for Sextic - Hamilton, detailed sketch by Klein Research Goal - Understand precise geometric relationship between solutions of Hamilton and Klein Hopefully gives insight to resolvent degree, as well. Start with Klein’s solution of the quintic. May 16 th , 2019 Alex Sutherland 45 / 74

  65. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Remaining Goals for the Talk Formulas for Sextic - Hamilton, detailed sketch by Klein Research Goal - Understand precise geometric relationship between solutions of Hamilton and Klein Hopefully gives insight to resolvent degree, as well. Start with Klein’s solution of the quintic. May 16 th , 2019 Alex Sutherland 45 / 74

  66. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Formula for the Quintic Theorem (Klein) P 1 E Φ 5 E 2 I P 1 P 1 E √△ 5 φ √− P 1 P 1 E 1 △ 5 √− A C 5 P 1 is a formula for the quintic (in one variable functions). May 16 th , 2019 Alex Sutherland 46 / 74

  67. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Components of The Tower • E 1 � C 5 - reduction of quintic to the normal form z 5 + az 2 + bz + c • E √△ 5 � E 1 - adjoin square root of discriminant • Icosahedral cover I : P 1 � P 1 ∼ = P 1 /A 5 [ H ( z 1 , z 2 ) 3 : 1728 f ( z 1 , z 2 ) 5 ] [ z 1 : z 2 ] �� f, H - polynomials invariant under action of A 5 (correspond to vertices, faces) May 16 th , 2019 Alex Sutherland 47 / 74

  68. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Components of The Tower • E 1 � C 5 - reduction of quintic to the normal form z 5 + az 2 + bz + c • E √△ 5 � E 1 - adjoin square root of discriminant • Icosahedral cover I : P 1 � P 1 ∼ = P 1 /A 5 [ H ( z 1 , z 2 ) 3 : 1728 f ( z 1 , z 2 ) 5 ] [ z 1 : z 2 ] �� f, H - polynomials invariant under action of A 5 (correspond to vertices, faces) May 16 th , 2019 Alex Sutherland 47 / 74

  69. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Components of The Tower • E 1 � C 5 - reduction of quintic to the normal form z 5 + az 2 + bz + c • E √△ 5 � E 1 - adjoin square root of discriminant • Icosahedral cover I : P 1 � P 1 ∼ = P 1 /A 5 [ H ( z 1 , z 2 ) 3 : 1728 f ( z 1 , z 2 ) 5 ] [ z 1 : z 2 ] �� f, H - polynomials invariant under action of A 5 (correspond to vertices, faces) May 16 th , 2019 Alex Sutherland 47 / 74

  70. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic Components of The Tower • E 1 � C 5 - reduction of quintic to the normal form z 5 + az 2 + bz + c • E √△ 5 � E 1 - adjoin square root of discriminant • Icosahedral cover I : P 1 � P 1 ∼ = P 1 /A 5 [ H ( z 1 , z 2 ) 3 : 1728 f ( z 1 , z 2 ) 5 ] [ z 1 : z 2 ] �� f, H - polynomials invariant under action of A 5 (correspond to vertices, faces) May 16 th , 2019 Alex Sutherland 47 / 74

  71. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic The Analytic Part Klein - complete algebraic solution of the quintic Further, use analytic functions to solve polynomials. Example: z n = w 1 n log( w ) ⇔ z = e May 16 th , 2019 Alex Sutherland 48 / 74

  72. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic The Analytic Part Klein - complete algebraic solution of the quintic Further, use analytic functions to solve polynomials. Example: z n = w 1 n log( w ) ⇔ z = e May 16 th , 2019 Alex Sutherland 48 / 74

  73. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic The Analytic Part Klein - complete algebraic solution of the quintic Further, use analytic functions to solve polynomials. Example: z n = w 1 n log( w ) ⇔ z = e May 16 th , 2019 Alex Sutherland 48 / 74

  74. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic The Analytic Part P 1 uniformized by upper half-plane H For quintic, use elliptic modular functions. May 16 th , 2019 Alex Sutherland 49 / 74

  75. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic The Analytic Part P 1 uniformized by upper half-plane H For quintic, use elliptic modular functions. May 16 th , 2019 Alex Sutherland 49 / 74

  76. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic The Bring Curve Bring (1786) also gave a solution to the quintic Bring reduced generic quintic to z 5 + az + b If z 1 , . . . , z 5 are roots of a polynomial of the form z 5 + az + b , then 5 5 5 ∑ ∑ ∑ z 2 z 3 z k = k = k = 0 k =1 k =1 k =1 Equations define a subvariety C B ⊆ P 4 - Bring curve May 16 th , 2019 Alex Sutherland 50 / 74

  77. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic The Bring Curve Bring (1786) also gave a solution to the quintic Bring reduced generic quintic to z 5 + az + b If z 1 , . . . , z 5 are roots of a polynomial of the form z 5 + az + b , then 5 5 5 ∑ ∑ ∑ z 2 z 3 z k = k = k = 0 k =1 k =1 k =1 Equations define a subvariety C B ⊆ P 4 - Bring curve May 16 th , 2019 Alex Sutherland 50 / 74

  78. Solving Polynomials and Resolvent Degree Introduction Algebraic Functions and Formulas Resolvent Degree Klein’s Solution to the Quintic The Sextic The Bring Curve Bring (1786) also gave a solution to the quintic Bring reduced generic quintic to z 5 + az + b If z 1 , . . . , z 5 are roots of a polynomial of the form z 5 + az + b , then 5 5 5 ∑ ∑ ∑ z 2 z 3 z k = k = k = 0 k =1 k =1 k =1 Equations define a subvariety C B ⊆ P 4 - Bring curve May 16 th , 2019 Alex Sutherland 50 / 74

Recommend

The Perrin-McClintock Resolvent, Solvable Quintics and Plethysms Frank Grosshans In his seminal

The Perrin-McClintock Resolvent, Solvable Quintics and Plethysms Frank Grosshans In his seminal

The Perrin-McClintock Resolvent, Solvable Quintics and Plethysms Frank Grosshans In his seminal paper of 1771, Lagrange found that certain polynomials of degree 6 called resolvents could be used to determine whether a quintic polynomial was

873 views • 33 slides
A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Kim

A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Kim

A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Kim Batselier KU Leuven Department of Electrical Engineering

1.14k views • 109 slides
A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Kim

A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Kim

A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Kim Batselier KU Leuven Department of Electrical Engineering

1.19k views • 108 slides
Problem solving and search C h a p t e r 3 (Adapted from Stuart Russel, Dan Klein, and others.

Problem solving and search C h a p t e r 3 (Adapted from Stuart Russel, Dan Klein, and others.

Problem solving and search C h a p t e r 3 (Adapted from Stuart Russel, Dan Klein, and others. Thanks!) Chapter 3 1 Outline Problem-solving agents Problem types Problem formulation Example problems Basic search algorithms

576 views • 35 slides
The Power of Low-Degree Polynomials for Solving Statistical Problems Alex Wein Courant

The Power of Low-Degree Polynomials for Solving Statistical Problems Alex Wein Courant

The Power of Low-Degree Polynomials for Solving Statistical Problems Alex Wein Courant Institute, New York University Based on joint work with: David Gamarnik (MIT) Aukosh Jagannath (Waterloo) Tselil Schramm (Stanford) 1 / 10 Problems in

835 views • 51 slides
Draft EE 8235: Lecture 9 1 Lecture 9: Spectral theory for compact normal operators Resolvent

Draft EE 8235: Lecture 9 1 Lecture 9: Spectral theory for compact normal operators Resolvent

Draft EE 8235: Lecture 9 1 Lecture 9: Spectral theory for compact normal operators Resolvent and spectrum of an operator Compact operators Direct extension of matrices Normal operators Commute with its adjoint Compact

396 views • 10 slides
Randomness in C 2 and Pluripotential Theory Randomness in C 2 and Pluripotential Theory Outline 1

Randomness in C 2 and Pluripotential Theory Randomness in C 2 and Pluripotential Theory Outline 1

Randomness in C 2 and Pluripotential Theory Randomness in C 2 and Pluripotential Theory Outline 1 Zeros of univariate random polynomials p : C C and potential theory; recent results of Bloom-Dauvergne 2 Random polynomials p : C 2 C and

585 views • 42 slides
Advances in Knot Polynomials 21 October 2016 Advances in Knot Polynomials 21 October 2016 1 /

Advances in Knot Polynomials 21 October 2016 Advances in Knot Polynomials 21 October 2016 1 /

Advances in Knot Polynomials 21 October 2016 Advances in Knot Polynomials 21 October 2016 1 / 49 ABSTRACT Review of achievements and problems in the theory of colored knot polynomials. Accent is on the current mystery around the differential

761 views • 49 slides
Back to the Roots Polynomial System Solving Using Linear Algebra Philippe Dreesen KU Leuven

Back to the Roots Polynomial System Solving Using Linear Algebra Philippe Dreesen KU Leuven

Motivation and History Univariate Polynomials Multivariate Polynomials Algebraic Optimization Conclusions Back to the Roots Polynomial System Solving Using Linear Algebra Philippe Dreesen KU Leuven Department of Electrical Engineering

795 views • 42 slides
Correlation bounds for polynomials, and the disproof of a conjecture on Gowers' norm using Ramsey

Correlation bounds for polynomials, and the disproof of a conjecture on Gowers' norm using Ramsey

Correlation bounds for polynomials, and the disproof of a conjecture on Gowers' norm using Ramsey theory Emanuele Viola Northeastern University May 2011 Polynomials Polynomials: degree d, n variables over F 2 = {0,1} E.g., p = x 1 + x 5 +

411 views • 17 slides
Endpoint resolvent estimates for compact Riemannian manifolds joint work with R. L. Frank to

Endpoint resolvent estimates for compact Riemannian manifolds joint work with R. L. Frank to

Endpoint resolvent estimates for compact Riemannian manifolds joint work with R. L. Frank to appear in J. Funct. Anal. (arXiv:1611.00462) Lukas Schimmer California Institute of Technology 13 February 2017 Schimmer (Caltech) Endpoint resolvent

274 views • 16 slides
Orthogonal Polynomials for Bernoulli and Euler Polynomials Lin JIU Dalhousie University Number

Orthogonal Polynomials for Bernoulli and Euler Polynomials Lin JIU Dalhousie University Number

Orthogonal Polynomials for Bernoulli and Euler Polynomials Lin JIU Dalhousie University Number Theory Seminar Jan. 14th, 2019 Acknowledgment Diane Shi Tianjin University Objects Bernoulli numbers B n : Euler numbers E n :

598 views • 32 slides
Fields and Polynomials (a 3 ,b 3 ) (a 2 ,b 2 ) (a 5 ,b 5 ) b 1 (a 4 ,b 4 ) a 1 First, a little

Fields and Polynomials (a 3 ,b 3 ) (a 2 ,b 2 ) (a 5 ,b 5 ) b 1 (a 4 ,b 4 ) a 1 First, a little

15-251: Great Theoretical Ideas in Computer Science Fall 2016 Lecture 24 November 17, 2016 Fields and Polynomials (a 3 ,b 3 ) (a 2 ,b 2 ) (a 5 ,b 5 ) b 1 (a 4 ,b 4 ) a 1 First, a little more Number Theory Bezouts identity Let a,b be

1.07k views • 72 slides
Legendre Polynomials Bernd Schr oder logo1 Bernd Schr oder Louisiana Tech University,

Legendre Polynomials Bernd Schr oder logo1 Bernd Schr oder Louisiana Tech University,

Overview Solving the Legendre Equation Application Legendre Polynomials Bernd Schr oder logo1 Bernd Schr oder Louisiana Tech University, College of Engineering and Science Legendre Polynomials Overview Solving the Legendre Equation

942 views • 82 slides
Solving problems in topological groups (and number theory) using category theory G abor Luk

Solving problems in topological groups (and number theory) using category theory G abor Luk

Solving problems in topological groups (and number theory) using category theory G abor Luk acs dr.gabor.lukacs@gmail.com suspended without pay from University of Manitoba Winnipeg, Manitoba, Canada Category Theory Oktoberfest, October

328 views • 14 slides
Polynomials, Number Theory, and Experimental Mathematics Michael Mossinghoff Davidson

Polynomials, Number Theory, and Experimental Mathematics Michael Mossinghoff Davidson

Challenges in 21st Century Experimental Mathematical Computation Polynomials, Number Theory, and Experimental Mathematics Michael Mossinghoff Davidson College ICERM Brown University July 21-25, 2014 Overview Three

1.04k views • 41 slides
Introduction to D -module Theory. Algorithms for Computing Bernstein-Sato Polynomials Jorge

Introduction to D -module Theory. Algorithms for Computing Bernstein-Sato Polynomials Jorge

Introduction to D -module Theory. Algorithms for Computing Bernstein-Sato Polynomials Jorge Mart n-Morales Centro Universitario de la Defensa de Zaragoza Academia General Militar Differential Algebra and Related Topics October 27-30,

866 views • 42 slides
WARD KLEIN WARD KLEIN WARD KLEIN WARD KLEIN ti Chi f E Forward-Looking Statements The

WARD KLEIN WARD KLEIN WARD KLEIN WARD KLEIN ti Chi f E Forward-Looking Statements The

Chief Executive Officer Offi WARD KLEIN WARD KLEIN WARD KLEIN WARD KLEIN ti Chi f E Forward-Looking Statements The following presentation contains forward looking statements within the meaning of the Private Securities Litigation

1.18k views • 92 slides
Introduction to Zonal Polynomials Lin Jiu Dalhousie University Number Theory Seminar Jan. 22,

Introduction to Zonal Polynomials Lin Jiu Dalhousie University Number Theory Seminar Jan. 22,

Introduction to Zonal Polynomials Lin Jiu Dalhousie University Number Theory Seminar Jan. 22, 2018 Hypergeometric function and Pochhammer symbol z n a 1 , . . . , a s ( a 1 ) n ( a s ) n s F t : z := n ! , b 1 , .

1.17k views • 77 slides
High-precision threshold of the toric code from spin-glass theory and graph polynomials . . .

High-precision threshold of the toric code from spin-glass theory and graph polynomials . . .

. High-precision threshold of the toric code from spin-glass theory and graph polynomials . . . Masayuki Ohzeki Kyoto University 2015/07/01 This is in collaboration with Prof. Jesper L. Jacobsen (ENS). J. Phys. A: Math. Theor. 48 095001

345 views • 32 slides
A GENERALIZATION OF SYMANZIK POLYNOMIALS MASTER THESIS, UNDER THE SUPERVISION OF OMID AMINI

A GENERALIZATION OF SYMANZIK POLYNOMIALS MASTER THESIS, UNDER THE SUPERVISION OF OMID AMINI

A GENERALIZATION OF SYMANZIK POLYNOMIALS MASTER THESIS, UNDER THE SUPERVISION OF OMID AMINI MATTHIEU PIQUEREZ Abstract. Symanzik polynomials are defined on Feynman graphs. They are used in quan- tum field theory to compute Feynman amplitudes. But

952 views • 79 slides
Part II: Interpolation and Approximation theory Contents: Review of Lagrange interpolation

Part II: Interpolation and Approximation theory Contents: Review of Lagrange interpolation

Part II: Interpolation and Approximation theory Contents: Review of Lagrange interpolation polynomials 1 Newton interpolation 2 Optimal interpolation points; Chebyshev polynomials 3 Cubic spline interpolation 4 Error analysis 5 Numerical

1.14k views • 47 slides
Approximation algorithms for graph polynomials and partition functions. Guus Regts University of

Approximation algorithms for graph polynomials and partition functions. Guus Regts University of

Approximation algorithms for graph polynomials and partition functions. Guus Regts University of Amsterdam 14 June 2016, Dagstuhl Seminar Graph Polynomials: Towards a Comparative Theory Based on joint work with Viresh Patel (University of

454 views • 29 slides
Perturbation theory (Ch. 6) (time-independent, nondegenerate, Sec. 6.1) Usually solving TISE

Perturbation theory (Ch. 6) (time-independent, nondegenerate, Sec. 6.1) Usually solving TISE

EE201/MSE207 Lecture 15 Perturbation theory (Ch. 6) (time-independent, nondegenerate, Sec. 6.1) Usually solving TISE = is too complicated; need approximations. Perturbation theory is one of approximate methods to solve TISE.

766 views • 9 slides

More recommend