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

• Functional Transcendence via Groups and Galois Theories

Michael F. Singer

Department of Mathematics North Carolina State University

Raleigh, NC 27695-8205 singer@math.ncsu.edu

Functional Transcendence around Ax-Schanuel

Oxford, 30 September 2014

• Theorem: (Hölder, 1887) The Gamma function defined by

Γ(x + 1)− xΓ(x) = 0

satisfies no polynomial differential equation that is, there is no nonzero 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.

• Theorem: (Hölder, 1887) The Gamma function defined by Γ(x + 1)− xΓ(x) = 0 satisfies no polynomial differential equation.

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

u(x + 1)− b(x)u(x) = 0.

has a meromorphic solution that is differentially algebraic over C(x) if and only if there exists a nonzero homogeneous linear differential polynomial 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)???

• Group Theory

⇓ Galois Theory

Form of Functional Dependencies if they occur

• Galois Theory of Linear Differential Equations

Galois Theory of Linear Differential Equations with Continuous Parameters

Galois Theory of Linear Differential Equations with Discrete Parameters/Action

Galois Theory of Linear Difference Equations with Continuous or Discrete Parameters/Action

• Picard-Vessiot (PV) Theory

dY dx

= A(x)Y A ∈ Mn(C(x))

Galois group = the group of transformations of Y that preserve all algebraic 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 closed Galois Correspondence:

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

• Picard-Vessiot (PV) Theory - Examples

Ex. 1: dydx = 1

2x y

K = C(x), E = K (x 12 ), Gal(E/K ) = Z/2Z ⊂ GL1(C)

Ex. 2: dydx = t x y

Gal(E/K ) =

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

Ex. 3: d 2y

dx2 − x −1 dy

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

ν /∈ Z + 12

• E = PV-extension of C(x)

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

Ex. 3(bis): d 2y

dx2 − x −1 dy

dx + (1− ν 2x−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).

• 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 ∈ Z s.t. g

g = na.

• Galois Theory of Linear Differential Equations with Continuous Parameter

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

Galois group = the group of transformations of Y that preserve all algebraic 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

• 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 uu ⇒ ∂t (

∂t u u ) = 0

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

) = 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 closed containing C(t). GLn(k) IS big enough.

• ∂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 all algebraic 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

• ∂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}

• 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 α 0 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 ).

• 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. The elements y1, . . . , yn are differentially dependent over K iff there is a linear differential operator w.r.t. ∂∂t L with coefficients in k and g ∈ K such that

L(a1, . . . ,an) = ∂xg.

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

0 s t−1e−sds

satisfies ∂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 ).

• Galois Theory of Linear Differential Equations with Discrete 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 d2y dx2

+ x dy dx

+ (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

d2y dx2 − 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).

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

d2y dx2

= 1

2x 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

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

K = δσ-field k = 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 = k

R = K [Y 1det Y ]σ is called the σ-PV ring.

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

• K = δσ-field k = K δ = {c ∈ K | δc = 0} E = K 〈Y 〉σ a σ-PV extension for σY = AY R = K [Y , 1det 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 by σ-Gal(E/K )