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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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