A New Approach for Formal Reduction of Singular Linear Differential - PowerPoint PPT Presentation

Introduction Previous methods The new approach Conclusion and prescriptive A New Approach for Formal Reduction of Singular Linear Differential Systems Using Eigenrings M.A. Barkatou, Joelle Saadé , J-A. Weil XLIM, Université de Limoges ISSAC, 16-19 July 2018, New York 1/23 M.A. Barkatou, Joelle Saadé , J-A. Weil Formal Reduction of Differential Systems, ISSAC 2018

Introduction Previous methods The new approach Conclusion and prescriptive Linear differential system Let A be a n × n matrix with coefficients over C (( x )) For clarity, C algebraically closed. [ A ] : Y ′ = A ( x ) Y , 1 where A ( x ) = x − q − 1 � ∞ i = 0 x i A i = x q + 1 ( A 0 + A 1 x + . . . ) . ◮ A i are constant square matrices of dimension n with A 0 � = 0. ◮ q is called the Poincaré rank of the system [ A ] . 2/23 M.A. Barkatou, Joelle Saadé , J-A. Weil Formal Reduction of Differential Systems, ISSAC 2018

Introduction Previous methods The new approach Conclusion and prescriptive Linear differential system Let A be a n × n matrix with coefficients over C (( x )) For clarity, C algebraically closed. [ A ] : Y ′ = A ( x ) Y , 1 where A ( x ) = x − q − 1 � ∞ i = 0 x i A i = x q + 1 ( A 0 + A 1 x + . . . ) . ◮ A i are constant square matrices of dimension n with A 0 � = 0. ◮ q is called the Poincaré rank of the system [ A ] . Hukahara[1937]-Turrittin[1955]-Levelt[1975] : A Formal Fundamental Matrix Solution (FFMS) can be written as Y ( x ) = φ ( x 1 / s ) x Λ exp ( Q ( x − 1 / s )) ◮ s is the global ramification. ◮ Q is the exponential part of [ A ] . 2/23 M.A. Barkatou, Joelle Saadé , J-A. Weil Formal Reduction of Differential Systems, ISSAC 2018

Introduction Previous methods The new approach Conclusion and prescriptive One strategy to compute an FFMS is to ◮ First compute the exponential part Q . ◮ Complete by applying algorithms for regular singular case ( Q = 0). 3/23 M.A. Barkatou, Joelle Saadé , J-A. Weil Formal Reduction of Differential Systems, ISSAC 2018

Introduction Previous methods The new approach Conclusion and prescriptive One strategy to compute an FFMS is to ◮ First compute the exponential part Q . ◮ Complete by applying algorithms for regular singular case ( Q = 0). Aim of the talk : New algorithm for computing Q . 3/23 M.A. Barkatou, Joelle Saadé , J-A. Weil Formal Reduction of Differential Systems, ISSAC 2018

Introduction Previous methods The new approach Conclusion and prescriptive One strategy to compute an FFMS is to ◮ First compute the exponential part Q . ◮ Complete by applying algorithms for regular singular case ( Q = 0). Aim of the talk : New algorithm for computing Q . One strategy to compute Q is to ◮ Compute an "equivalent system" with simple structure : Formal Reduction . 3/23 M.A. Barkatou, Joelle Saadé , J-A. Weil Formal Reduction of Differential Systems, ISSAC 2018

Introduction Previous methods The new approach Conclusion and prescriptive Formal Reduction Gauge Transformation : Change of variable Y = PZ , where P ∈ GL n ( C (( x ))) , leads to a system [ B ] : Z ′ = B ( x ) Z , B = P [ A ] := P − 1 AP − P − 1 P ′ Systems [ A ] and [ B ] are called equivalent. ( A C (( x )) B ) ∼ 4/23 M.A. Barkatou, Joelle Saadé , J-A. Weil Formal Reduction of Differential Systems, ISSAC 2018

Introduction Previous methods The new approach Conclusion and prescriptive Formal Reduction Gauge Transformation : Change of variable Y = PZ , where P ∈ GL n ( C (( x ))) , leads to a system [ B ] : Z ′ = B ( x ) Z , B = P [ A ] := P − 1 AP − P − 1 P ′ Systems [ A ] and [ B ] are called equivalent. ( A C (( x )) B ) ∼ ◮ Maximal decomposition : Ö B 1 0 0  è  ... [ A ] ∼  = [ B 1 ] ⊕ · · · ⊕ [ B ℓ ]  0 0   C (( x )) 0 0 B ℓ where each block B i is indecomposable over C (( x )) . 4/23 M.A. Barkatou, Joelle Saadé , J-A. Weil Formal Reduction of Differential Systems, ISSAC 2018

Introduction Previous methods The new approach Conclusion and prescriptive Formal Reduction Gauge Transformation : Change of variable Y = PZ , where P ∈ GL n ( C (( x ))) , leads to a system [ B ] : Z ′ = B ( x ) Z , B = P [ A ] := P − 1 AP − P − 1 P ′ Systems [ A ] and [ B ] are called equivalent. ( A C (( x )) B ) ∼ ◮ Maximal decomposition : Ö B 1 0 0  è  ... [ A ] ∼  = [ B 1 ] ⊕ · · · ⊕ [ B ℓ ]  0 0   C (( x )) 0 0 B ℓ where each block B i is indecomposable over C (( x )) . ◮ More refined decomposition requires field extensions i.e P ∈ GL n ( C (( x 1 / s ))) . 4/23 M.A. Barkatou, Joelle Saadé , J-A. Weil Formal Reduction of Differential Systems, ISSAC 2018

Introduction Previous methods The new approach Conclusion and prescriptive Previous methods 5/23 M.A. Barkatou, Joelle Saadé , J-A. Weil Formal Reduction of Differential Systems, ISSAC 2018

Introduction Previous algorithms Previous methods Barkatou’s algorithm 1997 The new approach Example Conclusion and prescriptive Previous algorithms Turrittin[1955], Wasow[1967], Chen[1990], Levelt[1991] Barkatou[1997]-Pflugel[2000] 1 Let A = x q + 1 ( A 0 + A 1 x + . . . ) 1 A 0 has distinct eigenvalues : apply splitting lemma. 6/23 M.A. Barkatou, Joelle Saadé , J-A. Weil Formal Reduction of Differential Systems, ISSAC 2018

Introduction Previous algorithms Previous methods Barkatou’s algorithm 1997 The new approach Example Conclusion and prescriptive Previous algorithms Turrittin[1955], Wasow[1967], Chen[1990], Levelt[1991] Barkatou[1997]-Pflugel[2000] 1 Let A = x q + 1 ( A 0 + A 1 x + . . . ) 1 A 0 has distinct eigenvalues : apply splitting lemma. a 2 A 0 has one eigenvalue a : update A ← A − x q + 1 I . Now A 0 nilpotent. 6/23 M.A. Barkatou, Joelle Saadé , J-A. Weil Formal Reduction of Differential Systems, ISSAC 2018

Introduction Previous algorithms Previous methods Barkatou’s algorithm 1997 The new approach Example Conclusion and prescriptive Previous algorithms Turrittin[1955], Wasow[1967], Chen[1990], Levelt[1991] Barkatou[1997]-Pflugel[2000] 1 Let A = x q + 1 ( A 0 + A 1 x + . . . ) 1 A 0 has distinct eigenvalues : apply splitting lemma. a 2 A 0 has one eigenvalue a : update A ← A − x q + 1 I . Now A 0 nilpotent. Different strategies with common goal : Find transformations and apply splitting lemma . 6/23 M.A. Barkatou, Joelle Saadé , J-A. Weil Formal Reduction of Differential Systems, ISSAC 2018

Introduction Previous algorithms Previous methods Barkatou’s algorithm 1997 The new approach Example Conclusion and prescriptive Previous algorithms Turrittin[1955], Wasow[1967], Chen[1990], Levelt[1991] Barkatou[1997]-Pflugel[2000] 1 Let A = x q + 1 ( A 0 + A 1 x + . . . ) 1 A 0 has distinct eigenvalues : apply splitting lemma. a 2 A 0 has one eigenvalue a : update A ← A − x q + 1 I . Now A 0 nilpotent. Different strategies with common goal : Find transformations and apply splitting lemma . 3 Arnold-Wasow Forms - Shearing transformation. 6/23 M.A. Barkatou, Joelle Saadé , J-A. Weil Formal Reduction of Differential Systems, ISSAC 2018

Introduction Previous algorithms Previous methods Barkatou’s algorithm 1997 The new approach Example Conclusion and prescriptive Computing the exponential part Q [Barkatou1997] 1 Let A = x q + 1 ( A 0 + A 1 x + . . . ) When A 0 is nilpotent : 3 Apply Moser rank reduction and iterate. 7/23 M.A. Barkatou, Joelle Saadé , J-A. Weil Formal Reduction of Differential Systems, ISSAC 2018

Introduction Previous algorithms Previous methods Barkatou’s algorithm 1997 The new approach Example Conclusion and prescriptive Computing the exponential part Q [Barkatou1997] 1 Let A = x q + 1 ( A 0 + A 1 x + . . . ) When A 0 is nilpotent : 3 Apply Moser rank reduction and iterate. 4 A 0 nilpotent + q is minimal : need to introduce ramification 7/23 M.A. Barkatou, Joelle Saadé , J-A. Weil Formal Reduction of Differential Systems, ISSAC 2018

Introduction Previous algorithms Previous methods Barkatou’s algorithm 1997 The new approach Example Conclusion and prescriptive Computing the exponential part Q [Barkatou1997] 1 Let A = x q + 1 ( A 0 + A 1 x + . . . ) When A 0 is nilpotent : 3 Apply Moser rank reduction and iterate. 4 A 0 nilpotent + q is minimal : need to introduce ramification ◮ Katz invariant algorithm. 7/23 M.A. Barkatou, Joelle Saadé , J-A. Weil Formal Reduction of Differential Systems, ISSAC 2018

Introduction Previous algorithms Previous methods Barkatou’s algorithm 1997 The new approach Example Conclusion and prescriptive Computing the exponential part Q [Barkatou1997] 1 Let A = x q + 1 ( A 0 + A 1 x + . . . ) When A 0 is nilpotent : 3 Apply Moser rank reduction and iterate. 4 A 0 nilpotent + q is minimal : need to introduce ramification ◮ Katz invariant algorithm. 5 Iterate on sub-blocks. The algorithm finishes when n = 1 or q = 0. 7/23 M.A. Barkatou, Joelle Saadé , J-A. Weil Formal Reduction of Differential Systems, ISSAC 2018

Introduction Previous algorithms Previous methods Barkatou’s algorithm 1997 The new approach Example Conclusion and prescriptive Example [Barkatou2010] á ë 0 0 x 0 − x 2 x 2 − x 2 A = 1 1 x 2 x 4 0 1 0 x 2 x 2 − x 2 0 ◮ Moser irreducible and A 0 is nilpotent. ◮ The Katz invariant κ = 8 / 3 [Barkatou97]. ◮ Ramification x = t 3 . ◮ Apply Moser-algorithm to 3 t 2 A ( t 3 ) . ◮ We get an equivalent system, where A 0 has 4 distinct eigenvalues. 8/23 M.A. Barkatou, Joelle Saadé , J-A. Weil Formal Reduction of Differential Systems, ISSAC 2018