Lind ependance dune famille de fonctions euleriennes par la th - PowerPoint PPT Presentation

L’ind´ ependance d’une famille de fonctions euleriennes par la th´ eorie Picard-Vessiot non commutative ui 0 , G.H.E. Duchamp 1 , 3 , V.C. B` V. Hoang Ngoc Minh 2 , 3 , Q.H. Ngˆ o 4 , K. Penson 5 0 Hue University of Sciences, 77 - Nguyen Hue street - Hue city, Vietnam. 1 Universit´ e Paris 13, 99 avenue Jean-Baptiste Cl´ ement, 93430 Villetaneuse, France. 2 Universit´ e Lille, 1 Place D´ eliot, 59024 Lille, France. 3 LIPN-UMR 7030, 99 avenue Jean-Baptiste Cl´ ement, 93430 Villetaneuse, France. 4 University of Hai Phong, 171, Phan Dang Luu, Kien An, Hai Phong, Viet Nam. 5 Universit´ e Paris 6, 975252 Paris Cedex 05, France. Journ´ ees nationales de calcul formel du 2 au 6 Mars 2020, Luminy

Outline 1. Eulerian functions and algebraic combinatorial aspects 1.1 Families of eulerian functions 1.2 Representative series (with coefficients in a ring) 1.3 Kleene stars of the plane and conc -characters 2. Noncommutative PV theory and independences via words 2.1 First step of noncommutative PV theory 2.2 Independences over differential field & differential ring 2.3 Extended regularization of divergent polyzetas by Newton-Girard formula

EULERIAN FUNCTIONS AND ALGEBRAIC COMBINATORIAL ASPECTS

Families of eulerian functions For any z ∈ C such that | z | < 1, let 1 � ζ ( k )( − z ) k / k � ζ ( kr )( − z r ) k / k , r ≥ 2 . ℓ 1 ( z ) := γ z − and ℓ r ( z ) := − k ≥ 2 k ≥ 1 Γ y r (1 + z ) := e − ℓ r ( z ) ∀ r ≥ 1 , and B y r ( a , b ) := Γ y r ( a )Γ y r ( b ) / Γ y r ( a + b ) . { ℓ r } r ≥ 1 and { e ℓ r } r ≥ 1 ∪ { 1 } are 2 C -linearly independent. For r ≥ 1, let ϑ = e 2 i π/ r . We have, for | z | < 1, r − 1 r − 1 � � � ζ ( kr )( − z r ) k / k = ℓ 1 ( ϑ j z ) = − log(Γ y 1 (1 + ϑ j z )). ℓ r ( z ) = − k ≥ 1 j =0 j =0 Taking the exponential and using Weierstrass factorization, we also have r − 1 r − 1 1 + ϑ j z 1 � � e − ϑ j z e ℓ r ( z ) = e γϑ j z � � � n . Γ y 1 (1 + ϑ j z ) = n j =0 j =0 n ≥ 1 Thus, ℓ r is holomorphic 3 on the open unit disc and e ℓ r (resp. e − ℓ r ) is entire (resp. meromorphic) admitting a countable set of isolated zeros (resp. poles) on the complex plan which is � r − 1 j =0 ϑ j N ≤− 1 , for r ≥ 1. 1. Γ y 1 and B y 1 are the classical gamma and beta eulerian functions. 2. Since ( ℓ r ) r ≥ 1 is triangular then ( ℓ r ) r ≥ 1 is C -linearly free. So is ( e ℓ r − 1) r ≥ 1 , being triangular, then ( e ℓ r ) r ≥ 1 is C -linearly free and free from 1. 3. ∀ r ≥ 2 , ζ (2) ≥ ζ ( r ) ≥ 1 : this proves that the radius of convergence of any the ℓ r is exactly one. In other words ℓ r is holomorphic on the open unit disc.

Notations ◮ A �X� (resp. A � �X� � ) denotes the set of polynomials (resp. formal series) with coefficients in the commutative ring A and over the alphabet X (which is Y := { y k } k ≥ 1 or X := { x 0 , . . . , x m } ) generating the free monoid ( X ∗ , 1 X ∗ ). ◮ H ⊔ ⊔ ( X ) := ( A �X� , conc , 1 X ∗ , ∆ ⊔ ⊔ , e ) and , e ) with 4 H ( Y ) := ( A � Y � , conc , 1 Y ∗ , ∆ ∀ x ∈ X , ∆ ⊔ ⊔ x = x ⊗ 1 X ∗ + 1 X ∗ ⊗ x , ∀ y i ∈ Y , ∆ y i = y i ⊗ 1 Y ∗ + 1 Y ∗ ⊗ y i + � k + l = i y k ⊗ y l . ◮ Considering A as the differential ring of holomorphic functions on a simply connected domain Ω, denoted by ( H (Ω) , ∂ ) and equipped 1 Ω as the neutral element, the differential ring ( H (Ω) � �X� � , d ) is defined, for S ∈ H (Ω) � �X� � , by � d S = ( ∂ � S | w � ) w ∈ H (Ω) � �X� � . w ∈X ∗ Const ( H (Ω)) = C . 1 Ω and Const ( H (Ω) � � ) = C . 1 Ω � �X� �X� � . 4. Or equivalently, for x , y ∈ X , y i , y j ∈ Y and u , v ∈ X ∗ (resp. Y ∗ ), ⊔ 1 X ∗ = 1 X ∗ ⊔ u ⊔ ⊔ u = u and xu ⊔ ⊔ yv = x ( u ⊔ ⊔ yv ) + y ( xu ⊔ ⊔ v ), 1 Y ∗ = 1 Y ∗ u u = u and x i u y j v = y i ( u y j v ) + y j ( y i u v ) + y i + j ( u v ), and ∆ conc w = � u 1 , u 2 ∈X ∗ , u 1 u 2 = u u 1 ⊗ u 2 .

Representative series and Sweedler’s dual Theorem (noncommutative rational series 5 ) Let S ∈ A � �X� � . The following assertions are equivalent 1. The series S belongs to 6 A rat � �X� � . 2. There exists a linear representation ( ν, µ, η ) (of rank n) for S with ν ∈ M 1 , n ( A ) , η ∈ M n , 1 ( A ) and a morphism of monoids µ : X ∗ → M n , n ( A ) s.t. S = � w ∈X ∗ ( νµ ( w ) η ) w. 3. The shifts 7 { S ⊳ w } w ∈X ∗ (resp. { w ⊲ S } w ∈X ∗ ) lie within a finitely generated shift-invariant A-module. Moreover, if A is a field K, previous assertions are equivalent to 4. There exists ( G i , D i ) i ∈ F finite s.t. ∆ conc ( S ) = � i ∈ F finite G i ⊗ D i . Hence, H ◦ ( K rat � ⊔ , 1 X ∗ , ∆ conc , e ) , ⊔ ( X ) = �X� � , ⊔ ⊔ ( resp. H ◦ ( Y ) = ( K rat � � Y � � , , 1 X ∗ , ∆ conc , e )) . 5. This form is a version over a ring of the form presented at JNCF 2019. � is the (algebraic) closure by { conc , + , ∗} of � 6. A rat � �X� A . X in A � �X� � . It is ⊔ . A rat � closed under � Y � � is also closed under . ⊔ 7. The left (resp. right ) shift of S by P is P ⊲ S (resp. S ⊳ P ) defined by, for w ∈ X ∗ , � P ⊲ S | w � = � S | wP � (resp. � S ⊳ P | w � = � S | Pw � ).

Kleene stars of the plane and conc -characters Theorem (rational exchangeable series 8 ) � be the set of (syntactically) exchangeable 9 series and Let A exc � �X� A rat exc � �X� � the set of series admitting a linear representation with commuting matrices (hence, exchangeable). Then 10 1. A rat � ⊂ A rat � exc � �X� �X� � ∩ A exc � �X� � . The equality holds when A is a field and, if X is finite then A rat ⊔ { A rat � exc � �X� � = � x � �} x ∈X . ⊔ 2. If A is a Q -algebra without zero divisors, { x ∗ } x ∈X (resp. { y ∗ } y ∈ Y ) ⊔ , 1 X ∗ ) (resp. are algebraically independent over ( A �X� , ⊔ , 1 Y ∗ ) ) within ( A rat � ⊔ , 1 X ∗ ) (resp. ( A � Y � , �X� � , ⊔ , 1 Y ∗ ) ). Moreover, x ∗ is a conc -character. ( A rat � � Y � � , � = { P (1 − xQ ) − 1 } P , Q ∈ A [ x ] and if 3. For any x ∈ X , one has A rat � � x � A = K is an algebraically closed field then one also has � = span K { ( ax ) ∗ ⊔ K rat � ⊔ K � x �| a ∈ K } . � x � 4. ∀ S ∈ K � �X� � , K being a field, � ∗ �� ∆ conc ( S ) = S ⊗ S , � S | 1 X ∗ � = 1 ⇐ ⇒ S = c x x with c x ∈ K . x ∈X 8. This form is a version over a ring of the form presented at JNCF 2019. � then ( ∀ u , v ∈ X ∗ )(( ∀ x ∈ X )( | u | x = | v | x ) ⇒ � S | u � = � S | v � ). 9. i.e. if S ∈ A exc � �X� � s.t. � S | 1 X ∗ � = 0. Then S ∗ = � n ≥ 0 S n , so called Kleene star of S . 10. Let S ∈ A � �X�

Triangular sub bialgebras of ( A rat � � X � � , ⊔ , 1 X ∗ , ∆ conc , e ) ⊔ Let ( ν, µ, η ) be a linear representation of R ∈ A rat � � X � � and L be the Lie algebra generated by { µ ( x ) } x ∈ X . Let M ( x ) := µ ( x ) x , for x ∈ X . Then R = ν M ( X ∗ ) η . If { µ ( x ) } x ∈ X are triangular then let D ( X ) (resp. N ( X )) be the diagonal (resp. nilpotent) letter matrix s.t. M ( X ) = D ( X ) + N ( X ) then M ( X ∗ ) = (( D ( X ∗ ) T ( X )) ∗ D ( X ∗ )). Moreover, if X = { x 0 , x 1 } then M ( X ∗ ) = ( M ( x ∗ 1 ) M ( x 0 )) ∗ M ( x ∗ 1 ) = ( M ( x ∗ 0 ) M ( x 1 )) ∗ M ( x ∗ 0 ). If A is an algabraically closed field, the modules generated by the following families are closed by conc , ⊔ and coproducts : ⊔ E k ∈ A rat � ( F 0 ) E 1 x 1 . . . E j x 1 E j +1 , where � x 0 � � , ( F 1 ) E 1 x 0 . . . E j x 0 E j +1 , where E k ∈ A rat � � x 1 � � , E k ∈ A rat ( F 2 ) E 1 x i 1 . . . E j x i j E j +1 , where exc � � X � � , x i k ∈ X . It follows then that 1. R is a linear combination of expressions in the form ( F 0 ) (resp. ( F 1 )) iff M ( x ∗ 1 ) M ( x 0 ) (resp. M ( x ∗ 0 ) M ( x 1 )) is nilpotent, 2. R is a linear combination of expressions in the form ( F 2 ) iff L is solvable. Thus, if R ∈ A rat exc � � X � � ⊔ ⊔ A � X � then L is nilpotent.

NONCOMMUTATIVE PV THEORY AND INDEPENDENCE VIA WORDS