Functional Transcendence via Groups andGalois Theories

Michael F. Singer

Department of MathematicsNorth Carolina State University

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

Functional Transcendence around Ax-Schanuel

Oxford, 30 September 2014

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.

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)???

Group Theory

⇓ Galois Theory

Form of Functional Dependenciesif they occur

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

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

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

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).

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.

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

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.

∂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

∂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 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 ).

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 ).

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).

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

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 .

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.

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

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).

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}.

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.

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.

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.

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).