Telescopers for Rational and Algebraic Functions via Residues - PowerPoint PPT Presentation

1 / 22 Telescopers for Rational and Algebraic Functions via Residues Shaoshi Chen Department of Mathematics North Carolina State University, Raleigh July 19, 2012 Joint with Manuel Kauers and Michael F. Singer Shaoshi Chen Telescopers and Residues

2 / 22 Outline ◮ Motivation: enumerating 3D Walks. ◮ Integrability problems: Given f ∈ K ( y , z ), decide whether f = D y ( g ) + D z ( h ) for some g , h ∈ K ( y , z ). ◮ Telescoping problems: Given f ∈ k ( x , y , z ), find L ∈ k ( x ) � D x � such that L ( x , D x )( f ) = D y ( g ) + D z ( h ) for some g , h ∈ k ( x , y , z ). Shaoshi Chen Telescopers and Residues

3 / 22 Enumerating 3D Walks The Rook moves in a straight line as below in first quadrant of the 3D space. R n : The number of different Rook walks from (0 , 0 , 0) to ( n , n , n ). Shaoshi Chen Telescopers and Residues

4 / 22 2D-diagonals f ( m , n ): the number of different Rook walks from (0 , 0) to ( m , n ). 1 � f ( m , n ) x m y n = F ( x , y ) = . y x 1 − 1 − x − 1 − y m , n ≥ 0 The diagonal of F ( x , y ) is � f ( n , n ) x n . diag( F ) := n ≥ 0 Notation: F an algebraically closed field of char zero (= Q , C , . . . ). Lemma: Let G := y − 1 · F ( y , x / y ) and L ( x , D x ) be a linear differential operator with coefficients in F ( x ). Then with H ∈ F ( x , y ) ⇒ L ( x , D x ) ( G ) = D y ( H ) L (diag( F )) = 0 � �� � Telescoper Shaoshi Chen Telescopers and Residues

5 / 22 Telescopers for Rational Functions: The Bivariate Case Let F ( x ) � D x � be the ring of linear differential operators in x with coefficients in F ( x ). Problem. For f ∈ F ( x , y ), find L ∈ F ( x ) � D x � such that L ( x , D x ) ( f ) = D y ( g ) for some g ∈ F ( x , y ) . � �� � Telescoper Simpler Problem. For h ∈ F ( x , y ), decide whether h = D y ( g ) for some g ∈ F ( x , y ) Answer. h = D y ( g ) iff res y ( h , β ) = 0 for any root β of the den( h ). Idea. To find L ∈ F ( x ) � D x � such that h = L ( f ) has only zero residues. Shaoshi Chen Telescopers and Residues

6 / 22 Telescoping via Residues: The Bivariate Rational Case Hermite Reduction. f = D y ( g 1 ) + A B , where deg y ( A ) < deg y ( B ) and B squarefree. Rothstein-Trager Resultant. R ( x , z ) := resultant y ( B , A − zD y ( B )). R ( x , res y ( A / B , β )) = 0 for any root β of B in F ( x ) . Theorem (Abel 1827). There exists L ∈ F ( x ) � D x � s.t. L ( γ ) = 0 for any root γ ∈ F ( x ) of R ( x , z ). L ( res y ( f , β )) = res y ( L ( f ) , β ) = 0 ( ∀ β ) ⇒ L ( f ) = D y ( g ) . Shaoshi Chen Telescopers and Residues

7 / 22 Telescopers for 2D Rook Walks For the 2D Rook walks, the rational function is ( − 1 + y )( − y + x ) f := y ( y − 2 x − 2 y 2 + 3 xy ) Resultant: The Rothstein-Trager Resultant is R ( x , z ) := ( − x + 2 zx )(40 z 2 x 2 + x − 2 x 2 + x 3 − 4 z 2 x − 36 z 2 x 3 ) So the residues of f w.r.t. y are respectively � � r 1 = 1 (9 x − 1) ( x − 1) (9 x − 1) ( x − 1) 2 , r 2 = , r 3 = − 18 x − 2 18 x − 2 Annihilators for residues: L 1 = D x and L 2 = L 3 = (9 x 2 − 10 x + 1) D x + (18 x − 14) Finally, the telescoper for f is L := (9 x 2 − 10 x + 1) D 2 x + (18 x − 14) D x . Shaoshi Chen Telescopers and Residues

8 / 22 Recurrences R ( n ): the number of different Rook walks from (0 , 0) to ( n , n ). Let S n be the shift operator defined by S n ( R ( n )) = R ( n + 1).   �  = 0 R ( n ) x n ⇒ L ( x , D x ) P ( n , S n )( R ( n )) = 0 . n ≥ 0 For the 2D Rook walks, we get the linear recurrence: R ( n + 2) = ( − 10 n − 14) R ( n + 1) + 9 nR ( n ) ( R (1) = 2 , R (2) = 14). n + 2 Running the recurrence, R ( n ) is as follows. 2 , 14 , 106 , 838 , 6802 , 56190 , 470010 , 3968310 , . . . OEIS:A051708 Shaoshi Chen Telescopers and Residues

9 / 22 Enumerating 3D Walks The Rook moves in 3-dimensional space. Question: How many different Rook walks from (0 , 0 , 0) to ( n , n , n )? Shaoshi Chen Telescopers and Residues

10 / 22 3D-diagonals f ( m , n , k ): the number of different Rook walks from (0 , 0 , 0) to ( m , n , k ). 1 � f ( m , n , k ) x m y n z k = F ( x , y , z ) = . y x z 1 − 1 − x − 1 − y − 1 − z m , n ≥ 0 The diagonal of F ( x , y , z ) is � f ( n , n , n ) x n . diag( F ) := n ≥ 0 F := ( yz ) − 1 · F ( y , z / y , x / z ) and L ( x , D x ) ∈ F ( x ) � D x � . Then Lemma: Let ˜ (˜ L ( x , D x ) F ) = D y ( G )+ D z ( H ) with G , H ∈ F ( x , y , z ) ⇒ L (diag( F )) = 0 . � �� � Telescoper Shaoshi Chen Telescopers and Residues

11 / 22 Telescoping Problems Telescopers for trivariate rational functions: Given f ∈ F ( x , y , z ), find L ∈ F ( x ) � D x � such that for some g , h ∈ F ( x , y , z ). L ( x , D x )( f ) = D y ( g ) + D z ( h ) Telescopers for bivariate algebraic functions: Given α ( x , y ) algebraic over F ( x , y ), find L ∈ F ( x ) � D x � such that L ( x , D x )( α ) = D y ( β ) for some algebraic β ( x , y ) over F ( x , y ). Goal: The two telescoping problems above are equivalent! Shaoshi Chen Telescopers and Residues

12 / 22 Integrability Problems Rational Integrability: Given f ( y , z ) ∈ E ( y , z ), decide for some g , h ∈ E ( y , z ). f = D y ( g ) + D z ( h ) If such g , h exist, we say that f is rational Integrable w.r.t. y and z . Algebraic Integrability: Given α ( y ) algebraic over E ( y ), decide α = D y ( β ) for some algebraic β over E ( y ). If such β exists, we say that α is algebraic Integrable w.r.t. y . Goal: The two Integrable problems above are equivalent! Shaoshi Chen Telescopers and Residues

13 / 22 Residues Definition. Let f ∈ F ( x , y )( z ). The residue of f at β i w.r.t. z , denoted by res z ( f , β i ), is the coefficient α i , 1 in n m i α i , j � � f = ( z − β i ) j , where α i , j , β i ∈ F ( x , y ) . i =1 j =1 Lemma. Let f ∈ F ( x , y )( z ) and β ∈ F ( x , y ). ◮ ∂ ( res z ( f , β )) = res z ( ∂ ( f ) , β ) with ∂ ∈ { D x , D y } . ◮ f = D z ( g ) ⇔ All residues of f w.r.t. z are zero. Remark. The second assertion is not true for algebraic functions!!! Shaoshi Chen Telescopers and Residues

14 / 22 Equivalence between Two Integrability Problems Theorem (Integrability). Let f = A / B ∈ F ( x )( y , z ). Then f = D y ( g ) + D z ( h ) ⇔ res z ( f , β ) = D y ( γ β ) for all β s.t. B ( β ) = 0. Example 1. Let f = ( x + y + z ) − 1 . Since res z ( f , − x − y ) = 1 = D y ( y ), f is rational Integrable w.r.t. y and z . In fact, � � � � x + y x + y f = D y + D z − . x + y + z x + y + z Example 2. Let f = ( xyz ) − 1 . Since res z ( f , 0) = ( xy ) − 1 is not algebraic integrable, f is not rational Integrable w.r.t. y and z . Shaoshi Chen Telescopers and Residues

15 / 22 Equivalence between Two Telescoping Problems Theorem (Telescoping). Let f ∈ F ( x , y , z ) and L ∈ F ( x ) � D x � . Then L ( x , D x ) is a telescoper for f w.r.t. y and z � L ( x , D x ) is a telescoper for every residue of f w.r.t. z Remark. L i ( x , D x )( α i ) = D y ( β i ) , 1 ≤ i ≤ n ⇓ L = LCLM( L 1 , L 2 , . . . , L n ) is a telescoper for all α i . Shaoshi Chen Telescopers and Residues

16 / 22 Differentials and Residues Let K = F ( x , y )( α ) where α is an algebraic function over F ( x , y ). Think of α ( x , y ) as a parameterized family of algebraic functions of y (with parameter x ). Differentials. Ω K / F ( x ) := { β dy | β ∈ K } . ◮ df = 0 for all f ∈ F ( x ) and D x ( β dy ) = D x ( β ) dy . Residues. Let P be a place of K (with no ramification). Then any β ∈ K has a P -adic expansion � a i t i , β = where ρ ∈ Z , a i ∈ F ( x ) and t ∈ K . i ≥ ρ The residues of β at P is a − 1 , denoted by res P ( β ). ◮ res P ( D x ( β )) = D x ( res P ( β )). Shaoshi Chen Telescopers and Residues

17 / 22 Differential Equations for Residues Let K = F ( x , y )( α ) and β = A / B with A ∈ F ( x )[ y , α ] and B ∈ F ( x )[ y ]. Let B ∗ be the squarefree part of B w.r.t. y . Theorem. There exists L ∈ F ( x ) � D x � such that all residues of L ( α ) are zero and deg D x ( L ) ≤ [ K : F ( x , y )] · deg y ( B ∗ ) . Definition. A differential ω ∈ Ω K / F ( x ) is of second kind if all residues of ω are zero. Lemma. ◮ If ω is exact i.e. ω = d ( β ), then ω is of second kind. ◮ Let Φ K / F ( x ) := { differentials of second kind } / { exact differentials } . Then dim F ( x ) (Φ K / F ( x ) ) = 2 · genus(K) . Shaoshi Chen Telescopers and Residues

18 / 22 Telescopers for Bivariate Algebraic Functions Algorithm. Given α ( x , y ) algebraic over F ( x , y ), do 1. Compute L 1 ∈ F ( x ) � D x � such that ω = L 1 ( α ) dy is of second kind. 2. Find a 0 , . . . , a 2 g ∈ F ( x ) with g := genus( K ) with K = F ( x , y )( α ), not all zero, such that a 2 g D 2 g x ( ω ) + · · · + a 0 ω = d ( β ) for some β ∈ K . Remark. If α ∈ F ( x , y ), Step 2 is not needed since g = 0. ◮ If ω is of second kind, so is D i x ( ω ) for all i ∈ N . ◮ Shaoshi Chen Telescopers and Residues