A density theorem for parameterized differential Galois theory

A density theorem for parameterized differential Galois theory Thomas Dreyfus University Paris 7 The Kolchin Seminar in Differential Algebra, 31/01/2014, New York.

In this talk, we are interested in the differential equations on the form: z Y (z, t) = A(z, t)y (z, t), (1) where t = t 1,..., t n is a parameter, A M m (O U ({z})) and O U ({z}) is a ring we will define later. We want to define a parameterized differential Galois group for (1) and, analogously to the density theorem of Ramis, give a set of topological generators.

1 Parameterized Hukuhara-Turrittin theorem 2 3 4 5

Let U be a polydisc of C n. Let M U be the field fo meromorphic function on U. ˆK U := M U ((z)). O U ({z}) := { f i (t)z i ˆK U t U, z fi (t)z i is a germ of meromorphic function}. K U := Frac(O U ({z})).

Proposition There exist U U and ν N, such that we have an invertible matrix solution of (1) of the form: where Remark F (z, t) := Ĥ(z, t)el(t) log(z) e Q(z,t), Ĥ(z, t) GL m ( ˆK U [z 1/ν ]). L(t) M m (M U ). Q(z, t) := Diag(q i (z, t)), with q i (z, t) z 1/ν M U [z 1/ν ]. If we restrict U, we may assume that U = U.

Let t := { t1,..., tn }. We still consider (1) with the fundamental solution F (z, t) given by the parameterized Hukuhara-Turrittin theorem. Let K U be the ( z, t )-differential field generated by K U and the entries of F (z, t). Proposition K U K U is a parameterized Picard-Vessiot extension of (1), i.e, the field of the z -constants of K U is M U.

( ) Let Aut t KU z K U be the group of field automorphism of K U that commutes with all the derivations and that let K U invariant. Let us consider the representation: ρ F : ( ) Aut t KU z K U GL m (M U ) ϕ F 1 ϕ(f ). Theorem The image of ρ F is a differential subgroup of GL m (M U ), i.e, there exist P 1,..., P k, t -differential polynomials in coefficients in M U, such that: (A i,j ) ρ F (Aut t z ) P 1 (A i,j ) = = P k (A i,j ) = 0.

We define the Kolchin topology as the topology of GL m (M U ) in which closed sets are zero sets of differential algebraic polynomials in coefficients in M U. Proposition Let G be a subgroup of Aut t z ( KU K U ). If K U G = KU, then G is dense for Kolchin topology in Aut t z ( KU K U ).

We still consider (1) with parameterized Picard-Vessiot extension K U K U and with Galois group Aut t z ( KU K U ). Definition We define ˆm Aut t z ( KU K U ) by: ˆm ˆKU = Id. ˆm(z α ) = e 2iπα z α. ˆm(log(z)) = log(z) + 2iπ. q z 1/ν M U [z 1/ν ], ˆm(e q ) = e ˆm(q).

Definition We define ( the) exponential torus as the subgroup of Aut t KU z K U of elements τ that satisfies: τ ˆKU = Id. τ(z α ) = z α. τ(log(z)) = log(z). There exists α τ, a character of z 1/ν M U [z 1/ν ], such that q z 1/ν M U [z 1/ν ], τ(e q ) = α τ (q)e q.

Example Let us consider z 2 z Y (z) + Y (z) = z. (2)

Example Let us consider z 2 z Y (z) + Y (z) = z. (2) The formal power series h(z) = n 0( 1) n n!z n+1 is solution of (2).

Example Let us consider z 2 z Y (z) + Y (z) = z. (2) Let d (2Z + 1)π. The following function is solution of (2) where f (ζ) = 1 1+ζ S d (h) = e id 0 ( f (ζ)e ζ ) dζ, z

Example Let us consider z 2 z Y (z) + Y (z) = z. (2) S d (h) is 1-Gevrey asymptotic to h: N Sd (h) ( 1) n n!z n+1 AN+1 (N + 1)! z N+1, n 0 where A > 0 is a constant.

Let d R and let k Q >0. ˆB k ( a n z n ) = L d 1 (f ) (z) = e id L d k := ρ k L d 1 ρ 1/k. 0 a n Γ(n/k) ζn. f (ζ) ( )dζ. ze ζ z

Let us consider h(z) C[[z]] solution of a linear differential equation in coefficients germs of meromorphic functions. There exist Σ R (set of singular directions), (κ 1,..., κ r ) ( Q >0) r, such that if d / Σ, S d (h) = L d κ r L d κ 1 ˆB κ1 ˆB κr (h), is a germ of analytic solution on the sector arg(z) ]d π/2κ r, d + π/2κ r [. Σ is finite modulo 2πZ. The map h S d (h) induces a morphism of differential field.

Let us consider system z Y (z) = A(z)Y (z) with coefficients that are germs of meromorphic functions. Let Ĥ(z)eL log(z) e Q(z) be the Hukuhara-Turrittin solution of z Y (z) = A(z)Y (z). There exist Σ R finite modulo 2π, ε > 0, such that if d / Σ, S d (Ĥ(z) ) e L log(z) e Q(z) is solution of z Y (z) = A(z)Y (z) with entries that are germs of meromorphic functions on the sector arg(z) ]d π/2ε, d + π/2ε[.

For d R, let St d GL m (C) such that: ( ) ( ) S d Ĥ(z) e L log(z) e Q(z) = S d + Ĥ(z) e L log(z) e Q(z) St d, where d π/2ε < d < d < d + < d + π/2ε and [d, d[ ]d, d + ] Σ =.

Let us consider (1). Let Ĥ(z, t)el(t) log(z) e Q(z,t) be parameterized Hukuhara-Turrittin solution. If we restrict U, we may assume that: There exist (d i (t)) continuous in t and finite modulo 2πZ, that satisfies d i (t) < d i+1 (t). (κ 1,i,j,..., κ r,i,j ) ( Q >0) r, such that for all d(t) continuous in t that does not interect Σ t := d i (t), we have an analytic solution of (1): ) S (Ĥ(z, d(t) t) e L(t) log(z) e Q(z,t) := ) L d(t) κ r,i,j L d(t) κ 1,i,j ˆB κ1,i,j ˆB κr,i,j (Ĥi,j (z, t) e L(t) log(z) e Q(z,t),.

For d(t) Σ t continuous in t, let t St d(t) GL m (M U ) such that for all t 0 U, St d(t 0) GL m (C), is the Stokes matrix in the direction t 0 of z Y (z, t 0 ) = A(z, t 0 )Y (z, t 0 ). Proposition St d(t) Aut t z ( KU K U ).

Let us consider (1). Let K U K U be the ( parameterized ) Picard-Vessiot extension and Aut t KU z K U be the Galois group. Theorem (D) The group generated by the (parameterized) monodromy matrix, the exponential torus and the Stokes matrices is dense for Kolchin topology in Aut t z ( KU K U ).

Let us consider z Y (z, t) = A(z, t)y (z, t) with A M m (M U (z)). We can define, MU (z) M ( U (z), parameterized ) Picard-Vessiot extension and Aut t MU z (z) M U (z) the Galois group. Theorem (D) The group generated by the (parameterized) monodromy matrix, the exponential torus and the Stokes matrices of all the singularities is dense for Kolchin topology in Aut t z ( ) MU (z) M U (z).

Definition ) Let t0 := z. Let A 0 M m (M U (z). We say that the linear differential equation t0 Y = A 0 Y ) is completely integrable if there exist A 1,..., A n M m (M U (z) such that, for all 0 i, j n, ti A j tj A i = A i A j A j A i.

Theorem (Cassidy/Singer 1 2, D 1 3) We have the equivalence: 1 t0 Y = A 0 Y is completely integrable. 2 The parameterized differential Galois group of t0 Y = A 0 Y is conjugated to an algebraic subgroup of GL m (C). 3 The topological generator of the parameterized differential Galois group of t0 Y = A 0 Y are conjugated to constant matrices.

Definition We say that k is a so-called universal t -field, if it has characteristic 0 and for any t -field k 0 k, t -finitely generated over Q, and any t -finitely generated extension k 1 of k 0, there is a t -differential k 0 -isomorphism of k 1 into k.

Theorem ( D, Mitschi/Singer) Let G be a differential subgroup of GL m (k). Then, G is the global parameterized differential Galois group of some equation having coefficients in k(z) if and only if G contains a finitely generated subgroup that is Kolchin-dense in G.

Cassidy, Phyllis J.; Singer, Michael F., Galois theory of parameterized differential equations and linear differential algebraic groups. Differential equations and quantum groups, 113-155, IRMA Lect. Math. Theor. Phys., 9, Eur. Math. Soc., Zürich, 2007. Dreyfus, Thomas, A density theorem for parameterized differential Galois theory. To appear in Pacific Journal of Mathematics. Mitschi, Claude; Singer, Michael F., Monodromy groups of parameterized linear differential equations with regular singularities. Bull. Lond. Math. Soc. 44 (2012), no. 5, 913-930.