Diagonals of rational functions Main Conference of Chaire J. Morlet Artin approximation and infinite dimensional geometry 27 mars 2015 Pierre Lairez TU Berlin . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . .. . . . . .
. . . . . . . . . . . . . . . . Diagonals: definitions and properties Binomial sums Computing diagonals Diagonals: definitions and properties Binomial sums . . . . . . . . . . . . . . . . . . . . . . . . Computing diagonals
. . . . . . . . . . . . . . . . Diagonals: definitions and properties Binomial sums . Diagonal of a power series Défjnition . Computing diagonals . . . . . . . . . . . . . . . . . . . . . . ∑ a i 1 ,..., i n x i 1 1 · · · x i n ▶ f = n ∈ Q ⟦ x 1 ,. . . , x n ⟧ i 1 ,..., i n ∈ N n ∑ a i ,..., i t i ▶ diag f def = i ⩾ 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 1 . . . . . . . Diagonals: definitions and properties Binomial sums Computing diagonals A combinatorial problem Counting rook paths 2 . 3 4 5 6 1 2 3 4 5 6 7 def Easy recurrence: . . . . . . . . . . . . . . . . . . . . . . . . . of a recurrence? . . . . . . (7 , 10) = nb. of rook paths from (0 , 0) to ( i , j ) a i , j y ∑ ∑ a i , j = a k , j + a i , k k < i k < j x What about a n , n ? asymptotic? existence (0 , 0)
. . . . . . . . . . . . . . . . . Diagonals: definitions and properties Binomial sums Computing diagonals Recurrence relations for rook paths . . . . . . . . . . . . . . . . . . . . . . . ▶ dimension 2 9 nu n − (14 + 10 n ) u n +1 + (2 + n ) u n +2 = 0 ▶ dimension 3 − 192 n 2 (1 + n )(88 + 35 n ) u n +(1 + n )(54864 + 100586 n + 59889 n 2 + 11305 n 3 ) u n +1 − (2 + n )(43362 + 63493 n + 30114 n 2 + 4655 n 3 ) u n +2 +2(2 + n )(3 + n ) 2 (53 + 35 n ) u n +3 = 0 ▶ dimension 4 5000 n 3 (1 + n ) 2 (2705080 + 3705334 n + 1884813 n 2 + 421590 n 3 + 34983 n 4 ) u n − (1 + n ) 2 (80002536960 + 282970075928 n + · · · + 6386508141 n 6 + 393838614 n 7 ) u n +1 +2(2 + n )(143370725280 + 500351938492 n + · · · + 2636030943 n 7 + 131501097 n 8 ) u n +2 − (3 + n ) 2 (26836974336 + 80191745800 n + 100381179794 n 2 + · · · + 44148546 n 7 ) u n +3 +2(3 + n ) 2 (4 + n ) 3 (497952 + 1060546 n + 829941 n 2 + 281658 n 3 + 34983 n 4 ) u n +4 = 0
. . . . . . . . . . . . . . . . Diagonals: definitions and properties Binomial sums Computing diagonals Difgerential equation for diagonals difgerential equation with polynomial coefgicients . . . . . . . . . . . . . . . . . . . . . . . . 1 ∑ ∑ ∑ a i , j x i y j = a i , j = a k , j + a i , k ⇒ y x 1 − 1 − x − i , j ⩾ 0 1 − y k < i k < j a n , n t n = diag � � 1 ∑ . � � y x 1 − 1 − x − n ⩾ 0 1 − y Theorem (Lipshitz 1988) — “diagonal ⇒ difgerentially finite” If R ∈ Q ( x 1 ,. . . , x n ) ∩ Q ⟦ x 1 ,. . . , x n ⟧ , then diag R satisfies a linear c r ( t ) y ( r ) + · · · + c 1 ( t ) y ′ + c 0 ( t ) y = 0 .
. . . . . . . . . . . . . . . . . Diagonals: definitions and properties Binomial sums Computing diagonals More properties of diagonals . . . . . . . . . . . . . . . . . . . . . . . Theorem (Furstenberg 1967) — “algebraic ⇒ diagonal” If f ( t ) = ∑ a n t n is an algebraic series (i.e. P ( t , f ( t )) = 0 for some P ∈ Q [ x , y ] ), then it is the diagonal of a rational power series. Theorem (Furstenberg 1967) — “diagonal ⇒ algebraic mod p ” If ∑ a n t n ∈ Q ⟦ t ⟧ is the diagonal of a rational power series, then it is an algebraic series modulo p for almost all prime p .
. . . . . . . . . . . . . . . . Diagonals: definitions and properties Binomial sums Computing diagonals Example is not algebraic. Besides, . . . . . . . . . . . . . . . . . . . . . . . . Algebricity modulo p ( ) (3 n )! 1 ∑ n ! 3 t n = diag f = 1 − x − y − z n ▶ f ≡ (1 + t ) − 1 mod 5 4 1 + 6 t + 6 t 2 ) − 1 ( 6 ▶ f ≡ mod 7 1 + 6 t + 2 t 2 + 8 t 3 ) − 1 ( ▶ f ≡ 10 mod 11 ▶ … ( ) 27 t 2 − t f ′′ + (54 t − 1) f ′ + 6 f = 0 .
. . . . . . . . . . . . . . . Diagonals: definitions and properties . Computing diagonals def I def I We check . Binomial sums . . . . . . . . . . . . . . . . . . . . . . . Proof of algebricity modulo p F q , the base field Slicing operators — For r ∈ Z , E r � a i t i � a qi + r t i and E r � a I x I � ∑ ∑ ∑ ∑ = = a q I +( r ,..., r ) x I � � � � i i ▶ diag ◦ E r = E r ◦ diag ; ▶ x i E r ( F ) = E r ( x q i F ) ; ▶ G ( x ) E r ( F ) = E r ( G ( x ) q F ) , because G ( x q ) = G ( x ) q , where x q = x q 1 ,. . . , x q n ; ▶ If f ( t ) = ∑ i a i t i , then ∑ t r ∑ a qi + r t qi f ( t ) = 0 ⩽ r < q i
. . . . . . . . . . . . . . . Diagonals: definitions and properties Binomial sums Computing diagonals . We check I def . . def . . . . . . . . . . . . . . . . . . . . . . I Proof of algebricity modulo p F q , the base field Slicing operators — For r ∈ Z , E r � a i t i � a qi + r t i and E r � a I x I � ∑ ∑ ∑ ∑ = = a q I +( r ,..., r ) x I � � � � i i ▶ diag ◦ E r = E r ◦ diag ; ▶ x i E r ( F ) = E r ( x q i F ) ; ▶ G ( x ) E r ( F ) = E r ( G ( x ) q F ) , because G ( x q ) = G ( x ) q , where x q = x q 1 ,. . . , x q n ; ▶ If f ( t ) = ∑ i a i t i , then q t r � a qi + r t i � ∑ ∑ f ( t ) = � � 0 ⩽ r < q i
. . . . . . . . . . . . . . . . Binomial sums Computing diagonals def I def I We check . Diagonals: definitions and properties . . . . . . . . . . . . . . . . . . . . . . . Proof of algebricity modulo p F q , the base field Slicing operators — For r ∈ Z , E r � a i t i � a qi + r t i and E r � a I x I � ∑ ∑ ∑ ∑ = = a q I +( r ,..., r ) x I � � � � i i ▶ diag ◦ E r = E r ◦ diag ; ▶ x i E r ( F ) = E r ( x q i F ) ; ▶ G ( x ) E r ( F ) = E r ( G ( x ) q F ) , because G ( x q ) = G ( x ) q , where x q = x q 1 ,. . . , x q n ; ▶ If f ( t ) = ∑ i a i t i , then ∑ t r E r ( f ) q f ( t ) = 0 ⩽ r < q
. . . . . . . . . . . . . . . . Binomial sums Computing diagonals diag F Proof. F F . Diagonals: definitions and properties . . . . . . . . . . . . . . . . . . . . . . . Proof of algebricity modulo p Let R = A F ∈ F q ( x ) , d = max ( deg A , deg F ) and the F q -vector space { } ) � � ( ( d + n )) ( P � deg P ⩽ d V = ⊂ F q ⟦ t ⟧ dim V ⩽ n 1. Operators E r stabilize V . ( P ) = diag ◦ E r � PF q − 1 � E r ◦ diag � � F q = diag � E r ( PF q − 1 ) � ∈ V � �
. . . . . . . . . . . . . . . . Diagonals: definitions and properties Binomial sums Computing diagonals . Proof. . . diag . . . . . . . . . . . . . . . . . . . . . F Proof of algebricity modulo p Let R = A F ∈ F q ( x ) , d = max ( deg A , deg F ) and the F q -vector space { } ) � � ( ( d + n )) ( P � deg P ⩽ d V = ⊂ F q ⟦ t ⟧ dim V ⩽ n 1. Operators E r stabilize V . 2. Let f 1 ,. . . , f s be a basis of V . There exist c ij ∈ F q [ t ] such that ∑ c ij f q ∀ i , f i = j . j ( ∑ ) q ∑ ∑ t r E r ( f i ) q = t r f i = b ij f j 0 ⩽ r < q 0 ⩽ r < q j ( ∑ ) ∑ f q b ij t r = j j 0 ⩽ r < q
. . . . . . . . . . . . . . . Diagonals: definitions and properties Binomial sums Computing diagonals diag F . Proof. Vect . . . . . . . . . . . . . . . . . . . . . . . . . Proof of algebricity modulo p Let R = A F ∈ F q ( x ) , d = max ( deg A , deg F ) and the F q -vector space { } ) � � ( ( d + n )) ( P � deg P ⩽ d V = ⊂ F q ⟦ t ⟧ dim V ⩽ n 1. Operators E r stabilize V . 2. Let f 1 ,. . . , f s be a basis of V . There exist c ij ∈ F q [ t ] such that ∑ c ij f q ∀ i , f i = j . j 3. All the elements of V are algebraic. ∑ ij f q 2 ∀ i , f q j c q i = , etc. j ∆( R ) q k � � { } � Thus, over F q ( t ) , Vect � 0 ⩽ k ⩽ s ⊂ { } � � � { } � � f q k f q s � 1 ⩽ i ⩽ s � 0 ⩽ k ⩽ s , 1 ⩽ i ⩽ s ⊂ Vect i i
. . . . . . . . . . . . . . . Diagonals: definitions and properties Binomial sums Computing diagonals Characterization of diagonals? Conjecture (Christol 1990) linear difgerential equation with polynomial coefgicients, then it is the diagonal of a rational power series. . . . . . . . . . . . . . . . . . . . . . . . . . “integer coefgicients + convergent + difg. finite ⇒ diagonal” If ∑ a n t n ∈ Z ⟦ t ⟧ , has radius of convergence > 0 , and satisfies a A hierarchy of power series — For f ∈ Q ⟦ t ⟧ , let N ( f ) be the minimum number of variables x 1 ,. . . , x N ( f ) such that f = diag R ( x 1 ,. . . , x N ( f ) ) , with R rational power series, if any. ▶ N ( f ) = 1 ⇔ f is rational ▶ N ( f ) = 2 ⇔ f is algebraic irrational (∑ ) (3 n )! n ! 3 t n = 3 ▶ N n ▶ Qvestion : Find a f such that 3 < N ( f ) < ∞ .
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