Direct problem in differential Galois theory€¦ · The Galois theory of linear di erential...

Direct problem in di�erential Galois theory

Lucia Di Vizio

UVSQ & CNRS

Ban�, September 21st, 2017

Joint work with M. Barkatou, T. Cluzeau and J.-A. Weil

Table of contents

The Galois theory of linear di�erential equations in 5 minutes

Notation and basic facts

Direct problem

Why the Lie algebra is easier to calculate ?

De�nition of reduced forms

Step 1 of the algorithm

Di�erential modules

The algorithm

Scheme of the algorithm

Characterization of reduced forms

Aparicio-Compoint-Weil theorem

Di�erential Galois theory

(k = C(x), ∂ = ddx )

∂(Y ) = AY , with A ∈ kn×n

∃ a Picard-Vessiot extension K/k

i.e., a di�erential �eld extension (K , ∂)/(k , ∂) s.t.

K∂ = C and ∃U ∈ Gln(K ), with ∂(U) = AU and

K = k(U).

Gal(A) = Aut∂(K/k) ↪→ GLn(C)

Important properties :

I G := Gal(A) is an algebraic group ⇒ g := Lie(G )I g ⊂ End(C-vector space of solutions)

(k = C(x), ∂ = ddx )

K = k(U).

(k = C(x), ∂ = ddx )

K = k(U).

(k = C(x), ∂ = ddx )

K = k(U).

Direct problem

direct problem the di�erential Galois group

I Compoint-Singer (1999), for reductive systems

I Hrushovski (2004), Feng (2015)

I van der Hoeven (2007)

characterization of the reduced forms of ∂(Y ) = AY

I Aparicio-Compoint-Weil (2013), for completely reducible

systems

I Dreyfus-Weil (in progress) for reductive systems

algorithm to calculate the Lie algebra of the di�erentialGalois group of an absolutely irreducible di�erential system

Joint work with M. Barkatou, T. Cluzeau, J.-A. Weil

Direct problem

systems

Direct problem

systems

Reduced forms

∂Y = AY , A = (ai ,j) ∈ C(x)n2

, group G , g = Lie(G ).

Wei-Normann decomposition of A =∑r

h=1 αhMh

I α1, . . . , αr is a C-basis of∑

i ,j=1,...,n Cai ,jI M1, . . . ,Mr ∈ Cn2

Lie(A) := the smallest alg. Lie algebra/C containingM1, . . . ,Mr

Proposition (Kolchin-Kovacic)

g ⊂ Lie(A) and

∃P ∈ GLn(C(x)) s.t. B = P ′P−1 + PAP−1 and

g = Lie(B)

∂Z = BZ is a reduced form of ∂Y = AY

Reduced forms

∂Y = AY , A = (ai ,j) ∈ C(x)n2

h=1 αhMh

i ,j=1,...,n Cai ,jI M1, . . . ,Mr ∈ Cn2

g ⊂ Lie(A) and

g = Lie(B)

Reduced forms

∂Y = AY , A = (ai ,j) ∈ C(x)n2

h=1 αhMh

i ,j=1,...,n Cai ,jI M1, . . . ,Mr ∈ Cn2

g ⊂ Lie(A) and

g = Lie(B)

Reduced forms

∂Y = AY , A = (ai ,j) ∈ C(x)n2

h=1 αhMh

i ,j=1,...,n Cai ,jI M1, . . . ,Mr ∈ Cn2

g ⊂ Lie(A) and

g = Lie(B)

Reduced forms

∂Y = AY , A = (ai ,j) ∈ C(x)n2

h=1 αhMh

i ,j=1,...,n Cai ,jI M1, . . . ,Mr ∈ Cn2

g ⊂ Lie(A) and

g = Lie(B)

The point of view of di�erential modules

(M,∇) di�erential k-module of dimension n∂Y = AY associated di�erential system, with G and g = Lie(G )

⇒ g→ End(V ), where V = (K ⊗kM)∇ (=vector space ofsolutions)

Hypotheses

1. (M,∇) absolutely irreducible⇒ M⊗kM∗ is a direct sum of

irreducibles

2. g ⊂ sln(C)⇒ g is semi-simple

The algorithm

1. DecomposingM⊗kM∗

2. Find a candidate gguess ⊂M⊗kM∗ for g⊗C C(x)

Hypotheses

irreducibles

The algorithm

Hypotheses

irreducibles

The algorithm

Hypotheses

irreducibles

The algorithm

Input : ∂Y = AY // 1) Decomposition ofM⊗kM∗

��2) Candidate gguess

��3) Find P and M1, . . .Mr

��

if fails

partial validation : g⊗C C(x) ⊂ gguess

if fails

��

de�nitive validation

(M,∇) di�erential k-module of dimension n∂Y = AY associated di�erential system

x0 ordinary point for the system

Théorème (Aparicio-Compoint-Weil 2013)

I ∂Y = AY is a reduced form ⇔ ∀ Constr(M) and

∀Φ rational solution of ∂Y = Constr(A)Y , Φ is a

constant vector.I ∃P ∈ GLn(k̄) s.t. ∂Z = P[A]Z is a reduced form

∀ Constr(M) and ∀Φ rational solution of

∂Y = Constr(A)Y , P sends Φ over Φ(x0).

constant vector.

I ∃P ∈ GLn(k̄) s.t. ∂Z = P[A]Z is a reduced form

constant vector.I ∃P ∈ GLn(k̄) s.t. ∂Z = P[A]Z is a reduced form

Direct problem in differential Galois theory€¦ · The Galois theory of linear di erential...

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