Using effective homological algebra for factoring and decomposing of - PowerPoint PPT Presentation

Using effective homological algebra for factoring and decomposing of linear functional systems Thomas Cluzeau & Alban Quadrat { Thomas.Cluzeau, Alban.Quadrat } @inria.fr INRIA Sophia Antipolis, CAFE Project, 2004 route des lucioles, BP 93, 06902 Sophia Antipolis cedex, France. www-sop.inria.fr/cafe/Alban.Quadrat/index.html S´ eminaire ALGO, INRIA Rocquencourt, 26/06/2006 Alban Quadrat Factoring and decomposing linear functional systems

Introduction: factorization and decomposition • Let L ( ∂ ) be a scalar ordinary or partial differential operator. • When is it possible to find L 1 ( ∂ ) and L 2 ( ∂ ) such that: L ( ∂ ) = L 2 ( ∂ ) L 1 ( ∂ )? • We note that L 1 ( ∂ ) y = 0 ⇒ L ( ∂ ) y = 0. • L ( ∂ ) y = 0 is equivalent to the cascade integration: L 1 ( ∂ ) y = z & L 2 ( ∂ ) z = 0 . • When is the integration of L ( ∂ ) y = 0 equivalent to: L 2 ( ∂ ) z = 0 & L 1 ( ∂ ) u = 0? ( L 1 X + Y L 2 = 1 ⇒ L 1 ( X z ) = z ⇒ y = u + X z ) Alban Quadrat Factoring and decomposing linear functional systems

Introduction: factorization and decomposition • Let us consider the first order ordinary differential system: E ( t ) ∈ k ( t ) p × p . ∂ y = E ( t ) y , ( ⋆ ) • When does it exist an invertible change of variables y = P ( t ) z , such that ( ⋆ ) ⇔ ∂ z = F ( t ) z , where F = − P − 1 ( ∂ P − E P ) is either of the form: � F 11 � � F 11 � F 12 0 F = or F = ? 0 0 F 22 F 22 Alban Quadrat Factoring and decomposing linear functional systems

Factorization: known cases Square differential systems: Beke’s algorithm (Beke1894, Schwarz89, Bronstein94, Tsar¨ ev94. . . ) Eigenring (Singer96, Giesbrecht98, Barkatou-Pfl¨ ugel98, Barkatou01 - ideas in Jacobson37. . . ) Square ( q -)difference systems (generalizations): Barkatou01, Bomboy01. . . Square D -finite partial differential systems (connections): Li-Schwarz-Tsar¨ ev03, Wu05. . . Same cases in positive characteristic and modular approaches: van der Put95, C.03, Giesbrecht-Zhang03, C.-van Hoeij04,06, Barkatou-C.-Weil05. . . Alban Quadrat Factoring and decomposing linear functional systems

General Setting What about general linear functional systems? • Example (Saint Venant equations): linearized model around the Riemann invariants (Dubois-Petit-Rouchon, ECC99): � y 1 ( t − 2 h ) + y 2 ( t ) − 2 ˙ y 3 ( t − h ) = 0 , y 1 ( t ) + y 2 ( t − 2 h ) − 2 ˙ y 3 ( t − h ) = 0 . � d � • Let D = R dt , δ and consider the system matrix: � � δ 2 − 2 δ d 1 dt ∈ D 2 × 3 . R = δ 2 − 2 δ d 1 dt Question: ∃ U ∈ GL 3 ( D ), V ∈ GL 2 ( D ) such that: � � α 1 0 0 V R U = , α 1 , α 2 , α 3 ∈ D ? 0 α 2 α 3 Alban Quadrat Factoring and decomposing linear functional systems

Outline • Type of systems: Partial differential/discrete/differential time-delay. . . linear systems (LFSs). • General topic: Algebraic study of linear functional systems (LFSs) coming from mathematical physics, engineering sciences. . . • Techniques: Module theory and homological algebra. • Applications: Equivalences of systems, Galois symmetries, quadratic first integrals/conservation laws, decoupling problem. . . • Implementation: package morphisms based on OreModules : http://wwwb.math.rwth-aachen.de/OreModules . Alban Quadrat Factoring and decomposing linear functional systems

General methodology 1 A linear system is defined by means of a matrix R with entries in a ring D of functional operators: R y = 0 . ( ⋆ ) 2 We associate a finitely presented left D -module M with ( ⋆ ). 3 A dictionary exists between the properties of ( ⋆ ) and M . 4 Homological algebra allows us to check properties of M . 5 Effective algebra (non-commutative Gr¨ obner/Janet bases) leads to constructive algorithms. 6 Implementation (Maple, Singular/Plural, Cocoa. . . ). Alban Quadrat Factoring and decomposing linear functional systems

I. Ore Module associated with a linear functional system Alban Quadrat Factoring and decomposing linear functional systems

Ore algebras Consider a ring A , an automorphism σ of A and a σ -derivation δ : δ ( a b ) = σ ( a ) δ ( b ) + δ ( a ) b . Definition: A non-commutative polynomial ring D = A [ ∂ ; σ, δ ] in ∂ is called skew if ∀ a ∈ A , ∂ a = σ ( a ) ∂ + δ ( a ) . Definition: Let us consider A = k , k [ x 1 , . . . , x n ] or k ( x 1 , . . . , x n ). The skew polynomial ring D = A [ ∂ 1 ; σ 1 , δ 1 ] . . . [ ∂ m ; σ m , δ m ] is called an Ore algebra if we have: � σ i δ j = δ j σ i , 1 ≤ i , j ≤ m , σ i ( ∂ j ) = ∂ j , δ i ( ∂ j ) = 0 , j < i . ⇒ D is generally a non-commutative polynomial ring. Alban Quadrat Factoring and decomposing linear functional systems

Examples of Ore algebras • Partial differential operators: A = k , k [ x 1 , . . . , x n ] , k ( x 1 , . . . , x n ) , � � � � ∂ ∂ D = A ∂ 1 ; id , . . . ∂ n ; id , , ∂ x 1 ∂ x n P = � 0 ≤| µ |≤ m a µ ( x ) ∂ µ ∈ D , ∂ µ = ∂ µ 1 1 . . . ∂ µ n n . • Shift operators: D = A [ ∂ ; σ, 0] , A = k , k [ n ] , k ( n ) , P = � m i =0 a i ( n ) ∂ i ∈ D , σ ( a )( n ) = a ( n + 1) . • Differential time-delay operators: � � ∂ 1 ; id , d D = A [ ∂ 2 ; σ, 0] , A = k , k [ t ] , k ( t ) , dt P = � 1 ∂ j 0 ≤ i + j ≤ m a ij ( t ) ∂ i 2 ∈ D . Alban Quadrat Factoring and decomposing linear functional systems

Exact sequences g → M ′′ is f • Definition : A sequence of D -morphisms M ′ − → M − said to be exact at M if we have: ker g = im f . → M ′ is a D -morphism, we then have the • Example : If f : M − following exact sequences: i ρ 1 0 − → coim f � M / ker f − → ker f − → M − → 0. j κ 2 0 − → M ′ → coker f � M ′ / im f − → im f − − → 0. i f κ 3 0 − → M ′ → ker f − → M − − → coker f − → 0. Alban Quadrat Factoring and decomposing linear functional systems

A left D -module M associated with R η = 0 • Let D be an Ore algebra, R ∈ D q × p and a left D -module F . • Let us consider ker F ( R . ) = { η ∈ F p | R η = 0 } . • As in number theory or algebraic geometry, we associate with the system ker F ( R . ) the finitely presented left D -module: M = D 1 × p / ( D 1 × q R ) . • Malgrange’s remark: applying the functor hom D ( ., F ) to the finite free resolution (exact sequence) . R π D 1 × q D 1 × p − → − → − → 0 , M λ = ( λ 1 , . . . , λ q ) �− → λ R we then obtain the exact sequence: R . π ⋆ F q F p ← − ← − hom D ( M , F ) ← − 0 . η = ( η 1 , . . . , η p ) T R η ← − � Alban Quadrat Factoring and decomposing linear functional systems

Example: Linearized Euler equations • The linearized Euler equations for an incompressible fluid can be defined by the system matrix   ∂ 1 ∂ 2 ∂ 3 0   ∂ t 0 0 ∂ 1    ∈ D 4 × 4 , R =  0 ∂ t 0 ∂ 2 0 0 ∂ t ∂ 3 � � � � � � � � ∂ ∂ ∂ ∂ t , id , ∂ where D = R ∂ 1 , id , ∂ 2 , id , ∂ 3 , id , . ∂ x 1 ∂ x 2 ∂ x 3 ∂ t • Let us consider the left D -module F = C ∞ (Ω) (Ω open convex subset of R 4 ) and the D -module: M = D 1 × 4 / ( D 1 × 4 R ) . The solutions of R y = 0 in F are in 1 − 1 correspondence with the morphisms from M to F , i.e., with the elements of: hom D ( M , F ) . Alban Quadrat Factoring and decomposing linear functional systems

II. Morphisms between Ore modules finitely presented by two matrices R and R ′ of functional operators Alban Quadrat Factoring and decomposing linear functional systems

Morphims of finitely presented modules • Let D be an Ore algebra of functional operators. • Let R ∈ D q × p , R ′ ∈ D q ′ × p ′ be two matrices. • Let us consider the finitely presented left D -modules: M ′ = D 1 × p ′ / ( D 1 × q ′ R ′ ) . M = D 1 × p / ( D 1 × q R ) , • We are interested in the abelian group hom D ( M , M ′ ) of D -morphisms from M to M ′ : . R π D 1 × q D 1 × p − → − → M − → 0 ↓ f . R ′ π ′ D 1 × q ′ D 1 × p ′ M ′ − − → − → − → 0 . Alban Quadrat Factoring and decomposing linear functional systems

Morphims of finitely presented modules • Let D be an Ore algebra of functional operators. • Let R ∈ D q × p , R ′ ∈ D q ′ × p ′ be two matrices. • We have the following commutative exact diagram: . R π D 1 × q D 1 × p − → − → − → 0 M ↓ . Q ↓ . P ↓ f . R ′ π ′ D 1 × q ′ D 1 × p ′ M ′ − − → − → − → 0 . ⇒ ∃ P ∈ D p × p ′ , Q ∈ D q × q ′ such that: ∃ f : M → M ′ ⇐ R P = Q R ′ . Moreover, we have f ( π ( λ )) = π ′ ( λ P ), for all λ ∈ D 1 × p . Alban Quadrat Factoring and decomposing linear functional systems

Eigenring: ∂ y = E y & ∂ z = F z E , F ∈ A p × p , R = ∂ I p − E , R ′ = ∂ I p − F . • D = A [ ∂ ; σ, δ ], . ( ∂ p I − E ) π D 1 × p D 1 × p 0 − → − − − − − − → − → − → 0 M ↓ . Q ↓ . P ↓ f . ( ∂ I p − F ) π ′ D 1 × p D 1 × p M ′ 0 − → − − − − − → − → − → 0 . � σ ( P ) = Q ∈ A p × p , ( ∂ I p − E ) P = Q ( ∂ I p − F ) ⇐ ⇒ δ ( P ) = E P − σ ( P ) F . If P ∈ A p × p is invertible, we then have: F = − σ ( P ) − 1 ( δ ( P ) − E P ) . Alban Quadrat Factoring and decomposing linear functional systems