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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 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 )