Functional Transcendence via Groups and Galois Theories · Galois Theory of Linear Differential...

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Functional Transcendence via Groups and Galois Theories Michael F. Singer Department of Mathematics North Carolina State University Raleigh, NC 27695-8205 [email protected] Functional Transcendence around Ax-Schanuel Oxford, 30 September 2014

Transcript of Functional Transcendence via Groups and Galois Theories · Galois Theory of Linear Differential...

Page 1: Functional Transcendence via Groups and Galois Theories · Galois Theory of Linear Differential Equations Galois Theory of Linear Differential Equations with Continuous Parameters

Functional Transcendence via Groups andGalois Theories

Michael F. Singer

Department of MathematicsNorth Carolina State University

Raleigh, NC [email protected]

Functional Transcendence around Ax-Schanuel

Oxford, 30 September 2014

Page 2: Functional Transcendence via Groups and Galois Theories · Galois Theory of Linear Differential Equations Galois Theory of Linear Differential Equations with Continuous Parameters

Theorem: (Holder, 1887) The Gamma function defined by

Γ(x + 1)− xΓ(x) = 0

satisfies no polynomial differential equation that is, there is nononzero polynomial

P(x , y , y ′, y ′′, . . . , y (n)) ∈ C[x , y , y ′, . . . , y (n), . . .]

such that P(x , Γ(x), Γ′(x), . . . Γ(n)(x)) = 0.

Page 3: Functional Transcendence via Groups and Galois Theories · Galois Theory of Linear Differential Equations Galois Theory of Linear Differential Equations with Continuous Parameters

Theorem: (Holder, 1887) The Gamma function defined byΓ(x + 1)− xΓ(x) = 0 satisfies no polynomial differential equation.

Theorem: (Hardouin; van der Put; H.-S.) Let b(x) ∈ C(x). Theequation

u(x + 1)− b(x)u(x) = 0.

has a meromorphic solution that is differentially algebraic over C(x) ifand only if there exists a nonzero homogeneous linear differentialpolynomial L(Y ) with coefficients in C such that

L(u′(x)

u(x)) = g(x)

for some g(x) ∈ C(x), and so

L(b′(x)

b(x)) = g(x + 1)− g(x).

Ex: For Γ(x), L( 1x ) = g(x + 1)− g(x)???

Page 4: Functional Transcendence via Groups and Galois Theories · Galois Theory of Linear Differential Equations Galois Theory of Linear Differential Equations with Continuous Parameters

Group Theory

⇓ Galois Theory

Form of Functional Dependenciesif they occur

Page 5: Functional Transcendence via Groups and Galois Theories · Galois Theory of Linear Differential Equations Galois Theory of Linear Differential Equations with Continuous Parameters

Galois Theory of Linear Differential Equations

Galois Theory of Linear Differential Equations with ContinuousParameters

Galois Theory of Linear Differential Equations with DiscreteParameters/Action

Galois Theory of Linear Difference Equations with Continuous orDiscrete Parameters/Action

Page 6: Functional Transcendence via Groups and Galois Theories · Galois Theory of Linear Differential Equations Galois Theory of Linear Differential Equations with Continuous Parameters

Picard-Vessiot (PV) Theory

dYdx

= A(x)Y A ∈ Mn(C(x))

Galois group = the group of transformations of Y that preserve allalgebraic relations among x ,Y and the derivatives of Y .

Formally: Y = (yi,j ), yi,j analytic near x = x0, det Y 6= 0,

K = C(x) ⊂ C(x)(y1,1, . . . , yn,n) = E , PV-Extension

E is closed under ∂ = ddx

Gal(E/K ) = {σ | σ = K -autom. of E , σ∂ = ∂σ}

∀σ ∈ Gal(E/K ), ∂(σY ) = A(σY )⇒ ∃ Cσ ∈ GLn(C) s.t. σY = Y · Cσ

Gal(E/K ) ⊂ GLn(C) is Zariski closedGalois Correspondence:

HZariski closed ⊂ Gal(E/K ) ⇔ FDiff. field, k⊂F⊂E

Page 7: Functional Transcendence via Groups and Galois Theories · Galois Theory of Linear Differential Equations Galois Theory of Linear Differential Equations with Continuous Parameters

Picard-Vessiot (PV) Theory - Examples

Ex. 1: dydx = 1

2x y

K = C(x), E = K (x12 ), Gal(E/K ) = Z/2Z ⊂ GL1(C)

Ex. 2: dydx = t

x y

Gal(E/K ) =

{Z/qZ if t = p/q, (p,q) = 1C∗ = GL1(C) if t /∈ Q

Ex. 3: d2ydx2 − x−1 dy

dx + (1− ν2x−2)y = 0 Gal(E/K ) = SL2(C)

ν /∈ Z + 12

Page 8: Functional Transcendence via Groups and Galois Theories · Galois Theory of Linear Differential Equations Galois Theory of Linear Differential Equations with Continuous Parameters

E = PV-extension of C(x)

dimC Gal(E/C(x)) = tr.deg.C(x)E

Ex. 3(bis): d2ydx2 − x−1 dy

dx + (1− ν2x−2)y = 0 Gal(E/K ) = SL2(C)

ν /∈ Z + 12

K = C(x)(Yν ,Y ′ν , Jν , J ′ν) dimC SL2(C) = 3⇒ tr. deg.C(x)E = 3

YνJ ′ν − Y ′νJν ∈ C(x) and Yν ,Y ′ν , Jν alg. indep. over C(x).

Page 9: Functional Transcendence via Groups and Galois Theories · Galois Theory of Linear Differential Equations Galois Theory of Linear Differential Equations with Continuous Parameters

K a ddx - differential field of functions meromorphic in domain G ⊂ C.

a1, . . . ,an ∈ K

Proposition:∫

a1, . . . ,∫

an are algebraically dependent over K iff

∃c1, . . . cn ∈ C and g ∈ K such that

g′ = c1a1 + . . .+ cnan.

In particular, for a ∈ K ,∫

a is algebraic over K iff ∃g ∈ K s.t g′ = a.

Proposition: e∫

a1 , . . . ,e∫

an are alg. dep. over K iff∃g ∈ K , m1, . . . ,mn ∈ Z not all zero s.t.

g′

g= m1a1 + . . .+ mnan.

In particular, for a ∈ K , e∫

a is algebraic over K iff ∃g ∈ K ,n 6= 0 ∈ Zs.t. g′

g = na.

Page 10: Functional Transcendence via Groups and Galois Theories · Galois Theory of Linear Differential Equations Galois Theory of Linear Differential Equations with Continuous Parameters

Galois Theory of Linear Differential Equations withContinuous Parameter

∂xY = A(t , x)Y A ∈ Mn(C(t , x)), ∂x = ∂∂x

Galois group = the group of transformations of Y that preserve allalgebraic relations among x ,Y , and the {∂x , ∂t}-derivatives of Y .

Formally: Y = (yi,j ), yi,j analytic near x = x0, t = t0, det Y 6= 0,

K = C(t , x) ⊂ C(t , x)(y1,1, . . . , yn,n, ∂ty1,1, . . . , ∂mt yi,j,...) = E

∂t -PV-Extension

E is closed under ∂x , ∂t . Note: (Yt )x = (Yx )t = AtY + AYt

∂t -Gal(E/K ) = {σ | σ = k -autom., σ∂ = ∂σ ∂ ∈ {∂x , ∂t}}

∀σ ∈ ∂t -Gal(E/K ), ∂x (σY ) = A(σY )⇒ ∃ Cσ ∈ GLn(C(t)) s.t. σY = Y · Cσ

Page 11: Functional Transcendence via Groups and Galois Theories · Galois Theory of Linear Differential Equations Galois Theory of Linear Differential Equations with Continuous Parameters

Ex.4: ∂y∂x = t

x y

K = C(t , x), E = K (y = x t , ∂y∂t = (log x)x t ) = K (x t , log x),

σ ∈ ∂t -Gal(E/K )⇒

{σ(x t ) = ux t ⇒ u ∈ C(t)σ(log x) = log x + ∂t u

u ⇒ ∂t (∂t uu ) = 0

∂t -Gal(E/K ) = {(u(t)) ∈ GL1(C(t)) | ∂t (∂tuu

) = 0}

= C∗

E∂t -Gal(E/K ) = K (log x) 6= K

PROBLEM: GLn(C(t)) is not big enough.

SOLUTION: Replace C(t) with k = ∂t -differentially closedcontaining C(t). GLn(k) IS big enough.

Page 12: Functional Transcendence via Groups and Galois Theories · Galois Theory of Linear Differential Equations Galois Theory of Linear Differential Equations with Continuous Parameters

∂t-Picard-Vessiot (∂t-PV) Theory

∂xY = A(x)Y A ∈ Mn(k(x)), ∂x = ∂∂x , C(t) ⊂ k = ∂t -diff. closed.

Galois group = the group of transformations of Y that preserve allalgebraic relations among x ,Y , and the {∂x , ∂t}-derivatives of Y .

Formally: Y = (yi,j ), det Y 6= 0, formal solution

K = k(x) ⊂ k(x)(y1,1, . . . , yn,n, ∂ty1,1, . . . , ∂mt yi,j,...) = E

∂t -PV-Extension

E is closed under ∂x , ∂t

∂t -Gal(E/K ) = {σ | σ = K -autom., σ∂ = ∂σ ∂ ∈ {∂x , ∂t}}

∀σ ∈ ∂t -Gal(E/K ), ∂x (σY ) = A(σY )⇒ ∃ Cσ ∈ GLn(k) s.t. σY = Y · Cσ

Page 13: Functional Transcendence via Groups and Galois Theories · Galois Theory of Linear Differential Equations Galois Theory of Linear Differential Equations with Continuous Parameters

∂t -Gal(E/K ) = {σ | σ = K -autom., σ∂ = ∂σ ∂ ∈ {∂x , ∂t}}

∀σ ∈ ∂t -Gal(E/K ), ∂x (σY ) = A(σY )⇒ ∃ Cσ ∈ GLn(k) s.t. σY = Y · Cσ

∂t -Gal(E/kK ) ⊂ GLn(k) is a Linear Differential Algebraic Group

i.e., a Kolchin closed subgroup of GLn(k)

∂t -Gal(E/K ) is Zariski dense in Gal.

Galois Correspondence:

HKolchin closed ⊂ ∂t -Gal(E/K ) ⇔ FDiff. field, K⊂F⊂E

Ex.4: ∂y∂x = t

x y

K = k(x), E = (log x)x t ) = K (x t , log x),

∂t -Gal(E/K ) = {(u(t)) ∈ GL1(k) | ∂t (∂t uu ) = 0}

= {(u(t)) ∈ GL1(k) | uutt − (ut )2 = 0}

Page 14: Functional Transcendence via Groups and Galois Theories · Galois Theory of Linear Differential Equations Galois Theory of Linear Differential Equations with Continuous Parameters

Linear Differential Algebraic Groups (LDAGs)

k - a ∂t -differential field.

G ⊂ GLn(k), entries satisfy algebraic DEs with respect to t

G ( Ga(k0) = (k ,+) = {(

1 α0 1

)| α ∈ k}

⇒ G = {α | L(α) = 0} where L ∈ k [ ∂∂t ]

G ( Gm(k) = (k∗,×) = GL1(k)

⇒ G =

{Z/nZ{α 6= 0 | L(∂t (α)

α ) = 0} where L ∈ k [ ∂∂t ]

G a Zariski dense subset of H(k), a simple algebraic group⇒ G is conjugate to H(k∂t ).

Page 15: Functional Transcendence via Groups and Galois Theories · Galois Theory of Linear Differential Equations Galois Theory of Linear Differential Equations with Continuous Parameters

Applications

K = {∂x , ∂t} − differential field

k = K ∂x = {c ∈ K | ∂x (c) = 0} differentially closed

a ∈ K .

Proposition: Let y1, . . . , yn satisfy ∂xy1 = a1, . . . , ∂xyn = an. Theelements y1, . . . , yn are differentially dependent over K iff there is alinear differential operator w.r.t. ∂

∂tL with coefficients in k and g ∈ K

such thatL(a1, . . . ,an) = ∂xg.

Ex.5: The incomplete Gamma function γ(t , x) =∫ x

0 st−1e−sdssatisfies

∂γ

∂x= x t−1e−x

and γ, ∂γ∂t ,∂2γ∂2t , . . . are alg. ind. over K = k(x , log x , x t−1e−x ).

Page 16: Functional Transcendence via Groups and Galois Theories · Galois Theory of Linear Differential Equations Galois Theory of Linear Differential Equations with Continuous Parameters

Galois Theory of Linear Differential Equations withDiscrete Parameter/Action

K = a field with a derivation δ and endomorphism σ, δσ = σδ.

Ex.6: K = C(x , α), δ = ddx , σ(α) = α + 1. The Bessel function Jα(x)

satisfies

x2 d2ydx2 + x

dydx

+ (x2 − α2)y = 0, and

xJα+2(x)− 2(α + 1)Jα+1(x) + xJα(x) = 0.

Ex.7: K = C(x), δ = ddx , σ(x) = x + 1.

If Ai(x) and Bi(x) are lin. indep. solns. of

d2ydx2 − xy = 0

then Ai(x),Bi(x), δ(Ai(x)) are σ-independent over k , i.e.,Ai(x),Bi(x), δ(Ai(x)),Ai(x + 1),Bi(x + 1), δ(Ai(x + 1)),Ai(x + 2), . . .are alg. ind. over C(x).

Page 17: Functional Transcendence via Groups and Galois Theories · Galois Theory of Linear Differential Equations Galois Theory of Linear Differential Equations with Continuous Parameters

Ex.8: K = C(x), δ = ddx , σ(x) = x + 1.

d2ydx2 =

12x

y

σ-PV Extension: E = C(x ,√

x ,√

x + 1,√

x + 2, . . .)

Naive σ-PV Galois group: G = {id , φ}, φ(√

x) = −√

x .

PROBLEM:

EG = C(x ,√

x√

x + 1, . . . ,√

x + i√

x + j , . . .) 6= C(x).

Cannot be fixed by going to a larger field containing K δ = C

Page 18: Functional Transcendence via Groups and Galois Theories · Galois Theory of Linear Differential Equations Galois Theory of Linear Differential Equations with Continuous Parameters

σ-PV Theory: A functorial approachWork of C. Hardouin, L. Di Vizio, M. Wibmer.

K = δσ-fieldk = K δ = {c ∈ K | δc = 0}

δY = AY , A ∈ gln(K ) (1)

Definition: A σ-PV extension for (1) is a δσ-field E s.t.1. ∃Y ∈ GLn(E) s.t. δY = AY and E = K 〈Y 〉σ = k(Y , σY , σ2Y . . .)

2. Eδ = K δ = kR = K [Y 1

det Y ]σ is called the σ-PV ring.

Theorem. If K δ is algebraically closed, then there exists a σ-PVextension for (1). If K δ is σ-closed and E1,E2 are σ-PV extensionsthen for some ` > 0 E1 and E2 are isomorphic as δσ` extensions of K .

Page 19: Functional Transcendence via Groups and Galois Theories · Galois Theory of Linear Differential Equations Galois Theory of Linear Differential Equations with Continuous Parameters

K = δσ-fieldk = K δ = {c ∈ K | δc = 0}E = K 〈Y 〉σ a σ-PV extension for σY = AYR = K [Y , 1

det Y ]σ, the σ-PV ring.

The σ-PV Galois Group σ-Gal(E/K ) is the functor

σ-Gal(E/K ) : k -σ-algebras −→ groups

given byσ-Gal(E/K )(S) := Autδσ(R ⊗k S/K ⊗k S)

for every k -σ-algebra S where the action of δ on S is trivial and σ isan endomorphism of S.

Theorem. σ-Gal(E/K ) is representable by a finitely σ-generatedk -σ-algebra, in fact by (R ⊗K R)δ.

Page 20: Functional Transcendence via Groups and Galois Theories · Galois Theory of Linear Differential Equations Galois Theory of Linear Differential Equations with Continuous Parameters

Ex.8(bis): K = C(x), δ = ddx , σ(x) = x + 1. K δ = C. δy = 1

2x y

σ-PV extension: E = C(x)〈√

x〉σ = C(x ,√

x ,√

x + 1, . . .) = R.

σ-PV Galois Group: Let G = σ-Gal(E/K )

Let SEQ = {(c(0), c(1), c(2), . . .) | c(i) ∈ C} =∏∞

i=1 C.For c ∈ SEQ, σ(c) := (c(1), c(2), . . .).k = C ⊂ SEQ via α ∈ C↔ (α, α, . . .) ∈ SEQ.

G(SEQ) = Autδσ(R ⊗C SEQ/K ⊗C SEQ) = {(±1,±1, . . .)} ⊂ GL1(S).

Note: R ∩ (R ⊗C SEQ)G(SEQ) = K

In general, G(S) = {g ∈ GL1(S) | g2 = 1}.G is represented by C{y}σ/(y2 − 1)σ

Page 21: Functional Transcendence via Groups and Galois Theories · Galois Theory of Linear Differential Equations Galois Theory of Linear Differential Equations with Continuous Parameters

σ-Algebraic GroupsA σ-algebraic group over a difference field k is a functor G

G : k -σ-algebras −→ groups

which is representable by a finitely σ-generated k -σ-algebra.

Ex.9: Any linear algebraic group can be considered as a σ-algebraicgroup.

Ex.10: G a σ-closed subgroup of Ga: There is a set L of linearhomogeneous σ-polynomials s.t.

G(S) = {g ∈ Ga(S) | L(g) = 0 for all L ∈ L}

Ex.11: H a σ-closed subgroup of Gm: There is a setM ofmultiplicative polynomials s.t.

H(S) = {g ∈ Gm(S) | M(g) = 1 for all M ∈M}

A multiplicative polynomial is a σ-polynomial that is a product ofmonomials σi (y).

Page 22: Functional Transcendence via Groups and Galois Theories · Galois Theory of Linear Differential Equations Galois Theory of Linear Differential Equations with Continuous Parameters

Galois Correspondence

K = δσ-fieldk = K δ = {c ∈ k | δc = 0}E = K 〈Y 〉σ a σ-PV extension for δY = AYR = K [Y , 1

det Y ]σ, the σ-PV ring.

Theorem: The σ-PV Galois group is a σ-algebraic group. The map

F 7→ σ-Gal(E/F )

is a bijection between the set of intermediate δσ-fields K ⊂ F ⊂ Eand σ-closed subgroups of σ-Gal(E/F ).

Ex.8(bis): K = C(x), δ = ∂∂x , σ(x) = x + 1. K δ = C. δy = 1

2x y

F = C(x ,√

x + 1,√

x + 2, . . .) ⊂ C(x ,√

x ,√

x + 1,√

x + 2, . . .)

⇒ σ-Gal(K/F )(S) = {g ∈ GL1(S) | g2 = 1, σ(g) = 1}.

Page 23: Functional Transcendence via Groups and Galois Theories · Galois Theory of Linear Differential Equations Galois Theory of Linear Differential Equations with Continuous Parameters

Application: Discrete Integrability

Definition: K a δσ-field, A ∈ gln(K ), d ∈ Z>0. We say

δY = AY

is σd -integrable if ∃B ∈ GLn(K ) such that the system

δY = AYσdY = BY

is compatible, i.e.δB + BA = σd (A)B.

Ex.6(bis): Bessel Equation is σ-integrable

x2δ2y + xδy + (x2 − α2)y = 0, and

xJα+2(x)− 2(α + 1)Jα+1(x) + xJα(x) = 0.

Page 24: Functional Transcendence via Groups and Galois Theories · Galois Theory of Linear Differential Equations Galois Theory of Linear Differential Equations with Continuous Parameters

K = C(x), δ = ddx , σ(x) = x + 1

δY = AY , A ∈ gln(K ) (2)

E = σ-PV extension. Assume that the (usual) PV-Galois group isSLn. Then

G = σ-Gal(E/C(x)) is Zariski dense in SLn.If σ-dim.C(x)E < n2 − 1 then G 6= SLn.If G is a proper σ-closed and Zariski-dense subgroup of SLn thenG is conjugate to a subgroup of SLσ

d

n = {g ∈ SLn | σd (g) = g}for some d ∈ Z>0.

If G is conjugate to a subgroup of SLσd

n then (2) is σd -integrable.

Theorem: If σ-dim.C(x)E < n2 − 1, then for some d ∈ Z>0, theequation

δB + BA = σd (A)B

has a solution B ∈ GLn(C(x)).

Airy Equation: Ai(x),Bi(x), δ(Ai(x)) are σ-independent.

Page 25: Functional Transcendence via Groups and Galois Theories · Galois Theory of Linear Differential Equations Galois Theory of Linear Differential Equations with Continuous Parameters

Galois Theory of Linear Difference Equations withParameters

Continuous Parameters: (Hardouin/Singer)

Theorem: (Holder, 1887) The Gamma function defined by

Γ(x + 1)− xΓ(x) = 0

satisfies no polynomial differential equation.

Theorem: (Ishizaki, 1998) If a(x),b(x) ∈ C(x) and z(x) /∈ C(x)satisfies

z(qx) = a(x)z(x) + b(x)

and is meromorphic on C(x), then z(x) is not differentially algebraicover G(x), where G is the field of q-periodic functions.

Also differential independence of solutions of certainq-hypergeometric functions.

Page 26: Functional Transcendence via Groups and Galois Theories · Galois Theory of Linear Differential Equations Galois Theory of Linear Differential Equations with Continuous Parameters

Discrete Parameters/Actions: (Ovchinnikov/Wibmer)

Theorem: The Gamma Function satisfies no difference equation withrespect to x 7→ x + c, c /∈ Q over M<1, the field of meromorphicfunctions f whose Nevanlinna characteristic satisfies T (f , r) = o(r).