Differential dependency of solutions of q-difference equations
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1 Differential dependency of solutions of q-difference equations Lucia Di Vizio (joint work with Charlotte Hardouin) Boston, January 5th, 2012
2 Difference systems and modules 1 Difference systems and modules 2 Quick survey of difference Galois theory 3 Hypertranscendence 4 Jacobi Theta function 5 Basis function field 6 Descending the Galois group from C E to C E 7 Rank 1 hypertranscendency result
3 Difference systems and modules (F, σ) = field equipped with a noncyclic automorphisms K = F σ = sub-field of σ-invariant elements of F Difference system σy = AY with A Gl ν (F ) Difference modules M = (M, Σ) over F M Σ : M M F -vectorial space of finite dimension ν σ-semilinear bijection Remark The horizontal vectors with respect to Σ solutions of a difference system.
4 Difference systems and modules If e is a basis of M, then A(x) GL ν (C(x)) s.t. Σ q e = ea(x): Σ q (ey (x)) = ey (x) Y (x) = A(x)Y (). Return
5 Quick survey of difference Galois theory Suppose K algebraically closed σ(y ) = AY, with A Gl ν (F ) (R, σ)/(f, σ) σ-simple ring generated by a fundamental matrix of solutions. Gal(M A ) = Aut σ (R/F ) GL ν (K) (van der Put-Singer) = { 1,..., n } set of derivations of F, commuting with each other and with σ K= -closure of K: F = Frac(F K) Gal (M A ) = Aut σ, ( R/ F ) GL ν ( K) (Hardouin-Singer) Problem. A too big field of definition. Descent results for Gal (M A ) to K: Peron-Nieto, Wibmer, Gillet-Gorchinskiy-Ovchinnikov, (DV-Hardouin).
6 Hypertranscendence Theorem (Hardouin-Singer) The differential dimension of Gal (M A ) is equal to the degree of differential transcendence (hypertranscendence) over F of the entrees of a fundamental solution matrix. Applications to: Hypertranscendency criteria for rang 1 differential equations Integrability criteria
7 Jacobi Theta function Jacobi Theta function θ q (x) = n Z q n(n 1)/2 x n, Solution of θ q () = θ q (x), Remark: l q (x) = xθ q θ q l q () = l q (x) + 1 We set: σ q (f (x)) = f (), δ x = x d dx, δ q = q d dq. σ q [δ x (l q )] = δ x (l q ) σ q [δ q (l q )] = δ q (l q ) (Heat equation) 2δ q θ q = δ 2 xθ q + δ x θ q
8 Jacobi Theta function δ = l q δ x + δ q. Then [δ, σ q ] = 0 (C E (x, l q ), σ q, δ) is a differntial-q-different field. Slightly refined version of Hardouin-Singer criterium applied to θ q : θ q is differentially algebraic iff a 1,..., a m C E and f C E (l q, x) s.t. ( ) i a iδ i δ() = σ q (f ) f.
9 Jacobi Theta function δ = l q δ x + δ q. Then [δ, σ q ] = 0 (C E (x, l q ), σ q, δ) is a differntial-q-different field. Slightly refined version of Hardouin-Singer criterium applied to θ q : θ q is differentially algebraic iff a 1,..., a m C E and f C E (l q, x) s.t. ( ) i a iδ i δ() = σ q (f ) f. Easy to verify!
10 Jacobi Theta function δ = l q δ x + δ q. Then [δ, σ q ] = 0 (C E (x, l q ), σ q, δ) is a differntial-q-different field. Slightly refined version of Hardouin-Singer criterium applied to θ q : θ q is differentially algebraic iff a 1,..., a m C E and f C E (l q, x) s.t. ( ) i a iδ i δ() = σ q (f ) f. Easy to verify! δ() ( ) l 2 q + l q = l q (x) + 1 = σ q l2 q + l q 2 2
11 Jacobi Theta function δ = l q δ x + δ q. Then [δ, σ q ] = 0 (C E (x, l q ), σ q, δ) is a differntial-q-different field. Slightly refined version of Hardouin-Singer criterium applied to θ q : θ q is differentially algebraic iff a 1,..., a m C E and f C E (l q, x) s.t. ( ) i a iδ i δ() = σ q (f ) f. Easy to verify! δ() ( ) l 2 q + l q = l q (x) + 1 = σ q l2 q + l q 2 2 combined with δ() ( ) δ(θq ) = σ q δ(θ q) θ q θ q
12 Jacobi Theta function δ = l q δ x + δ q. Then [δ, σ q ] = 0 (C E (x, l q ), σ q, δ) is a differntial-q-different field. Slightly refined version of Hardouin-Singer criterium applied to θ q : θ q is differentially algebraic iff a 1,..., a m C E and f C E (l q, x) s.t. ( ) i a iδ i δ() = σ q (f ) f. Easy to verify! δ() ( ) l 2 q + l q = l q (x) + 1 = σ q l2 q + l q 2 2 combined with gives δ() ( ) δ(θq ) = σ q δ(θ q) θ q θ q
13 Jacobi Theta function δ = l q δ x + δ q. Then [δ, σ q ] = 0 (C E (x, l q ), σ q, δ) is a differntial-q-different field. Slightly refined version of Hardouin-Singer criterium applied to θ q : θ q is differentially algebraic iff a 1,..., a m C E and f C E (l q, x) s.t. ( ) i a iδ i δ() = σ q (f ) f. Easy to verify! δ() combined with gives 2 δ(θ q) θ q ( ) l 2 q + l q = l q (x) + 1 = σ q l2 q + l q 2 2 δ() ( ) δ(θq ) = σ q δ(θ q) θ q θ q ) (l 2 q + l q = 2 δ q(θ q ) + l 2 q l q C E θ q
14 Jacobi Theta function 2 δ q(θ q ) θ q 2 δ q(θ q ) θ q = δ 2 xθ q + δ x θ q (heat eq.) + l 2 q l q = δ x (l q ) C E
15 Basis function field k= field of characteristic 0 k(q)= rational functions in q K= finite extension of k(q) = a fixed extension to K of the q 1 -adic norm of k(q) d R, d > 1, s.t. f (q) = d deg q (f ) f (q) k[q] (C, )/(K, )= smallest complete et algebraically closed extension of K Mer(C )= meromorphic functions on C := C {0} C E := Mer(C ) σq = fields of elliptic functions over C /q Z, δ q et δ x = x d dx act over Mer(C ) and { δx σ q = σ q δ x ; δ q σ q = σ q (δ x + δ q ).
16 Descending the Galois group from C E to C E C E = differential closure of C E with respect to δ x and δ = l q δ x + δ q Notice that: [δ x, σ q ] = [δ, σ q ] = 0 δ x δ δ δ x = δ x (l q (x))δ x with δ x (l q (x)) C E
17 Descending the Galois group from C E to C E Lemme There exists h C E of the order 1 differential equation over C E : [l q δ x + δ q ] (h) = δ x (l q (x))h, such that the derivations: { 1 = hδ x ; 2 = l q (x)δ x + δ q commutes with each other and with σ q. Remark C E (x, l q ) is a (σ q, )-field for = { 1, 2 }. C E (x, l q ) is also a (σ q, 2 )-field.
18 Descending the Galois group from C E to C E 2 2(θ q ) Gal 2 (θ q ) θ q { small group 2 -algebraicity ) (l 2 q + l q = 2 δ q(θ q ) 2 ( 2 (c) c ) } = 0 θ q + l 2 q l q C E
19 Rank 1 hypertranscendency result Theorem (Hardouin-Singer, DV-Hardouin) let u Mer(C ) a meromorphic solution y() = a(x)y(x), a(x) k(q, x). The following assertions are equivalent: 1 a(x) = µx r g() g(x), for some r Z, g k(q, x) and µ k(q). 2 u is solution of a nontrivial δ x -differential equation with coefficients in C E (x, l q ) (and therefore in C(x)). 3 u is solution of a nontrivial 2 -differential equation with coefficients in C E (x, l q ).
20 Rank 1 hypertranscendency result THANK YOU!
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