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Algebraic Diagonals and Walks Alin Bostan Louis Dumont Bruno Salvy INRIA, France July 8, 2015 Algebraic Diagonals and Walks Alin Bostan, Louis Dumont , Bruno Salvy

Diagonals Algebraic Diagonals and Walks Alin Bostan, Louis Dumont , Bruno Salvy

Diagonals of rational functions Definition � a i 1 ,..., i k x i 1 1 . . . x i k F ( x 1 , . . . , x k ) = k i 1 ,..., i k ≥ 0 ↓ � a n ,..., n x n Diag ( F )( x ) = n ≥ 0 Example: 1 1 1 1 1 � n + m � 1 � x n y m 1 2 3 4 5 1 − x − y = n 1 3 6 10 15 n , m ≥ 0 1 4 10 20 35 � � � 2 n � 1 � 1 5 15 35 70 x n Diag = 1 − x − y n n ≥ 0 Algebraic Diagonals and Walks Alin Bostan, Louis Dumont , Bruno Salvy

Applications of diagonals n � n � 2 � n + k � 2 n � n �� n + k � k � k � 3 � � � Number theory = k k k k j k = 0 k = 0 j = 0 Enumerative combinatorics Statistical physics Algebraic Diagonals and Walks Alin Bostan, Louis Dumont , Bruno Salvy

Univariate power series Alg: algebraic series Diag: diagonals of rational functions Alg ⊂ Diag: Furstenberg (1967) Diag ⊂ Diff. finite: Christol (1982), Lipshitz (1988) Algebraic: P ( x , f ( x )) = 0, P ∈ Q [ x , y ] Differentially finite: L ( x , ∂ x ) · f = 0, L ∈ Q ( x ) � ∂ x � Algebraic Diagonals and Walks Alin Bostan, Louis Dumont , Bruno Salvy

Univariate power series Alg: algebraic series Diag: diagonals of rational functions Quasi-alg: quasi-algebraic series Diag ⊂ Quasi-alg: Furstenberg (1967) Quasi-algebraic: algebraic modulo p for all primes p Algebraic Diagonals and Walks Alin Bostan, Louis Dumont , Bruno Salvy

Univariate power series Alg: algebraic series Diag: diagonals of rational functions Diag(2): diagonals of bivariate rational functions Quasi-alg: quasi-algebraic series Diag(2) = Alg: Furstenberg (1967) Quasi-algebraic: algebraic modulo p for all primes p Algebraic Diagonals and Walks Alin Bostan, Louis Dumont , Bruno Salvy

Diagonals of bivariate rational functions Theorem (Furstenberg, 1967) Algebraic univariate power series are exactly the diagonals of bivariate rational functions Example: � � � 2 n � 1 1 � x n = √ 1 − 4 x Diag = 1 − x − y n n ≥ 0 ( 1 − 4 x )∆ 2 − 1 = 0 Problem F ∈ Q ( x , y ) Compute an annihilating polynomial for Diag ( F ) Study the degree of algebraicity of Diag ( F ) Algebraic Diagonals and Walks Alin Bostan, Louis Dumont , Bruno Salvy

Our solution Turn a well-known formula into an algorithm: d � x � G ( x , y ) := 1 y − y i ( x ) + ∂ ρ i ( x ) � y , y = ∂ y ( . . . ) ∈ Q ( x )( y ) y F i = 1 small branch: lim x → 0 y i ( x ) = 0 y 1 , . . . , y c , y c + 1 , . . . , y d � �� � � �� � small branches large branches c � Diag ( F ) = ρ i ( x ) i = 1 Algebraic Diagonals and Walks Alin Bostan, Louis Dumont , Bruno Salvy

Our solution Turn a well-known formula into an algorithm: ρ 1 ( x ) , . . . , ρ d ( x ) : residues of G at y 1 ( x ) , . . . , y d ( x ) c � Diag ( F ) = ρ i ( x ) i = 1 Subproblem 1: polynomial annihilating the residues Compute the polynomial R = � d i = 1 ( y − ρ i ( x )) ∈ Q ( x )[ y ] . Subproblem 2: pure composed sum Compute the polynomial ( c ≤ d ) � ( y − ( ρ i 1 ( x ) + . . . + ρ i c ( x )) ∈ Q ( x )[ y ] Σ c R = i 1 <...< i c Algebraic Diagonals and Walks Alin Bostan, Louis Dumont , Bruno Salvy

Compute a polynomial cancelling the residues Subproblem 1: polynomial annihilating the residues Compute the polynomial R = � d i = 1 ( y − ρ i ( x )) ∈ Q ( x )[ y ] . y 1 ( x ) , . . . , y d ( x ) : distinct poles of G ( x , y ) ∈ Q ( x )( y ) ρ 1 ( x ) , . . . , ρ d ( x ) : residues of G at y 1 ( x ) , . . . , y d ( x ) 1 If y i is a simple pole, then ρ i ( x ) = P ( x , y i ( x )) . ∂ Q ∂ y ( x , y i ( x )) ρ i ( x ) is cancelled by the Rothstein-Trager resultant: � ∂ Q � ∂ y ( x , y ) z − P ( x , y ) , Q ( x , y ) Res z 2 if y i is a multiple pole: Bronstein resultants Algebraic Diagonals and Walks Alin Bostan, Louis Dumont , Bruno Salvy

Compute the pure composed sum of a polynomial Subproblem 2: pure composed sum Compute the polynomial ( c ≤ d ) � Σ c R = ( y − ( ρ i 1 ( x ) + . . . + ρ i c ( x )) ∈ Q ( x )[ y ] i 1 <...< i c Main tool: Newton sums. d ) y n � ( ρ n 1 + . . . + ρ n R ← → N ( R ) = n ! n ≥ 0 Strategy: R − →N ( R ) =: S ↓ Σ c R ← −N (Σ c R ) = Φ( S ( y ) , S ( 2 y ) , . . . , S ( cy )) Φ : polynomial Algebraic Diagonals and Walks Alin Bostan, Louis Dumont , Bruno Salvy

First result: algebraic data structure for diagonals d x , d y : degrees in x and y of the denominator of F Theorem (B., D., S., 2015) � d x + d y � Σ c R has degree at most in y (tight) d x "generically", Σ c R is irreducible over Q ( x ) "generically", Σ c R is computed in quasi-optimal time d x = d y = d 1 2 3 4 deg x Σ c R , deg y Σ c R (2, 2) (16, 6) (108, 20) (640, 70) The degree of algebraicity of the diagonal is generically exponential in the size of the input rational function. Algebraic Diagonals and Walks Alin Bostan, Louis Dumont , Bruno Salvy

Walks Algebraic Diagonals and Walks Alin Bostan, Louis Dumont , Bruno Salvy

Back to lattice walks S : set of steps d : amplitude of S Problem Compute the generating series of the bridges, excursions and meanders at precision N Naive algorithms: quadratic complexity O ( d 2 N 2 ) Algebraic Diagonals and Walks Alin Bostan, Louis Dumont , Bruno Salvy

A previously known linear time algorithm Theorem (Banderier, Flajolet, 2002) The generating series of the bridges, excursions and meanders are algebraic. Strategy: 1 Compute an algebraic equation for B , E , or M ; 2 Deduce a differential equation; O ( C d ) 3 Deduce a recurrence; 4 Compute initial conditions using a naive method; 5 Unroll the recurrence. Complexity of the expansion: O ( d 2 N ) (linear in N ) Complexity of the pre-processing: O ( C d ) (exponential in d ) Algebraic Diagonals and Walks Alin Bostan, Louis Dumont , Bruno Salvy

A new quasi-linear time algorithm with small pre-processing Fundamental fact: the generating series of the bridges is a diagonal Theorem (B., Chen, Chyzak, Li (2010)) Diagonals of bivariate rational functions satisfy polynomial-size differential equations. Strategy: 1 Directly compute a differential equation for the bridges; 2 Deduce the excursions from the formula B ( x ) = 1 + xE ′ ( x ) / E ( x ) Complexity of the expansion: ˜ O ( d 2 N ) (quasi-linear in N ) Complexity of the pre-processing: ˜ O ( d 5 ) (polynomial in d ) The meanders can be computed similarly. Algebraic Diagonals and Walks Alin Bostan, Louis Dumont , Bruno Salvy

Summary picture Algebraic Diagonals and Walks Alin Bostan, Louis Dumont , Bruno Salvy