Diagonals: Combinatorics, Asymptotics and Computer Algebra
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1 Diagonals: Combinatorics, Asymptotics and Computer Algebra Bruno Salvy Inria & ENS de Lyon
2 Families of Generating Functions (a n ) 7! A(z) := X n 0 a n z n counts the number of objects of size n captures some structure If (a n ) then A(z) is. counts the number of words of length n in a regular language. in a nonambiguous context-free language. satisfies a linear recurrence with polynomial coefficients rational algebraic (satisfies a polynomial equation P(z,A)=0) differentially finite (satisfies a LDE with polynomial coefficients) hundreds of examples 1/25
3 Univariate Generating Functions D-FINITE Aim: asymptotics of the coefficients, automatically. DIAGONAL ALGEBRAIC RATIONAL Def diagonal: later. 2/25
4 I. A Quick Review of Analytic Combinatorics in One Variable Principle: Dominant singularity exponential behaviour local behaviour subexponential terms
5 Coefficients of Rational Functions F 1 =1= 1 2 i I 1 dz 1 z z 2 z 2 a n = 1 2 i Z f(z) z n+1 dz = = As n increases, the smallest singularities dominate. F n = n n /25
6 Conway s sequence 1,11,21,1211,111221, Generating function for lengths: f(z)=p(z)/q(z) with deg Q=72. Smallest singularity: δ(f) ρ=1/δ(f) l n ρ n Fast univariate resolution: Sagraloff-Mehlhorn16 ρ Res(f,δ(f)) remainder exponentially small 4/25
7 Singularity Analysis Ex: Rational Functions A 3-Step Method: 1. Locate dominant singularities a. singularities; b. dominant ones 2. Compute local behaviour 3. Translate into asymptotics (1 z) log k 1 1 z 7! n 1 ( ) logk n, ( 62 N? ) 17z 3 9z 2 7z Numerical resolution with sufficient precision + algebraic manipulations 2. Local expansion (easy). 3. Easy. Useful property [Pringsheim Borel] a n 0 for all n real positive dominant singularity. 5/25
8 Algebraic Generating Functions P (z,y(z)) = 0 1a. Location of possible singularities Implicit Function Theorem: 1b. Analytic continuation finds the dominant ones: not so easy [FlSe NoteVII.36]. 2. Local behaviour (Puiseux): 3. Translation: easy. P (z,y(z)) = 0 (1 z), ( 2 Q) Numerical resolution with sufficient precision + algebraic manipulations 6/25
9 Differentially-Finite Generating Functions a n (z)y (n) (z)+ + a 0 (z)y(z) =0, a i polynomials 1a. Location of possible singularities. Cauchy-Lipshitz Theorem: 1b. Analytic continuation finds the dominant ones: only numerical in general. Sage code exists [Mezzarobba2016]. 2. Local behaviour at regular singular points: 3. Translation: easy. a n (z) =0 (1 z) log k 1, ( 2 Q,k 2 N) 1 z Numerical resolution with sufficient precision + algebraic manipulations 7/25
10 Example: Apéry s Sequences a n = nx k=0 2 2 n n + k X n, b n = a n k k k=1 1 n k 3 + X k=1 kx m=1 ( 1) m n 2m 3 k n m 2 n+k k n+m m 2 and c n = b n (3)a n have generating functions that satisfy vanishes at 0, = p = p 2 ' 0.03, 2 ' 34. z 2 (z 2 34z + 1)y (z 5)y =0 In the neighborhood of, all solutions behave like analytic µ p z(1 + ( z)analytic). Mezzarobba s code gives µ a ' 4.55, µ b ' 5.46, µ c ' 0. Slightly more work gives µ c =0, then c n and eventually, a proof that (3) is irrational. n [Apéry1978] 8/25
11 II. Diagonals
12 If F (z) = G(z) H(z) its diagonal is Definition is a multivariate rational function with Taylor expansion F (z) = X i2n n c i z i, F (t) = X k2n c k,k,...,k t k. in this talk 1 k +1 2k k 2k k : : 1 1 x y =1+x + y +2xy + x2 + y x 2 y x (1 x y)(1 x) =1+y+1xy x2 +y x 2 y 2 + Apéry s a k : 1 1 t(1 + x)(1 + y)(1 + z)(1 + y + z + yz + xyz) =1+ +5xyzt + 9/25
13 Multiple Binomial Sums Ex. (from A=B) S = X r 0 X n ( 1) n+r+s r s 0 n s n + s n + r 2n r s s r n Def.: expressions obtained from: n, n 7! n k n 7! C n, (n, k) 7! using +, x, multiplication by constants, affine changes of indices and indefinite summation: nx (m,n) 7! u m,k. k=0 (Kronecker) Thm. Diagonals univariate binomial sums. > BinomSums[sumtores](S,u): ( ) [Bostan-Lairez-S.17] 1 1 t(1 + u 1 )(1 + u 2 )(1 u 1 u 3 )(1 u 2 u 3 ) 10/25
14 Diagonals are Differentially Finite [Christol84,Lipshitz88] Thm. If F has degree d in n variables, ΔF satisfies a LDE with order d n, coeffs of degree d O(n). + algo in ops. Õ(d 8n ) D-finite diag. alg. rat. Compares well with creative telescoping when both apply. Christol s conjecture: All differentially finite power series with integer coefficients and radius of convergence in (0, ) are diagonals. [Bostan-Lairez-S.13,Lairez16] 11/25
15 Asymptotics Thm. [Katz70,André00,Garoufalidis09] a 0 +a 1 z+ D-finite, a i in z, radius in (0, ), then its singular points are regular with rational exponents X a n n n log k 1 (n) f,,k. n (,,k)2 finite set in Q Q N Ex. The number a n of walks from the origin taking n steps {N,S,E,W,NW} and staying in the first quadrant behaves like C n n with 62 Q not D-finite. = 1+ [Bostan-Raschel-S.14] arccos(u), 8u3 8u 2 +6u 1=0, u > 0. 12/25
16 Bivariate Diagonals are Algebraic [Pólya21,Furstenberg67] D-finite diag. alg. = diag 2 [Bostan-Dumont-S.17] rat. Thm. F=A(x,y)/B(x,y), deg d in x and y, then ΔF cancels a polynomial of degree 4 d in y and t. x satisfies 1 x 2 y t y t y t t y t y 6 t t t y t t y 4 54 t t y t t y 2 t 3 50 t t t y t t =0 + quasi-optimal algorithm. the differential equation is often better. 13/25
17 III. Analytic Combinatorics in Several Variables Here, we restrict to rational diagonals and simple cases
18 Starting Point: Cauchy s Formula If f = X i 1,...,i n 0 c i1,...,i z i 1 n 1 z i n n neighborhood of 0, then is convergent in the c i1,...,i n = 1 2 i n Z T f(z 1,...,z n ) dz 1 dz n z i 1+1 z i n+1 1 n for any small torus T ( z j = re i j ) around 0. Asymptotics: deform the torus to pass where the integral concentrates asymptotically. 14/25
19 Coefficients of Diagonals F (z) = G(z) H(z) c k,...,k = 1 2 i n Z T G(z) H(z) dz 1 dz n (z 1 z n ) k+1 rank Critical points: minimize z 1 z n on V = {z @z 1 z n 1 z n n Minimal ones: on the boundary of the domain of convergence of F (z).! apple 1 i.e. 1 = = n A 3-step method 1a. locate the critical points (algebraic condition); 1b. find the minimal ones (semi-algebraic condition); 2. translate (easy in simple cases). 15/25
20 Ex.: Central Binomial Coefficients 2k k : 1 1 x y =1+x + y +2xy + x2 + y x 2 y 2 + (1). Critical points: 1 x y =0,x= y =) x = y =1/2. (2). Minimal ones. Easy. In general, this is the difficult step. (3). Analysis close to the minimal critical point: a k = 1 ZZ (2 i) 2 Z 4k+1 2 i 1 1 x y dx dy (xy) k i exp(4(k + 1)(x 1/2) 2 ) dx 4k p k. Z dx (x(1 x)) k+1 residue saddle-point approx 16/25
21 Kronecker Representation for the Critical Points Algebraic part: ``compute the solutions of the system H(z) =0 1 = = n If Under genericity assumptions, a probabilistic algorithm running in bit ops finds: History and Background: see Castro, Pardo, Hägele, and Morais (2001) [Giusti-Lecerf-S.01;Schost02;SafeySchost16] System reduced to a univariate polynomial. 17/25
22 Example (Lattice Path Model) The number of walks from the origin taking steps {NW,NE,SE,SW} and staying in the first quadrant is F, F (x, y, t) = (1 + x)(1 + y) 1 t(1 + x 2 + y 2 + x 2 y 2 ) Kronecker representation of the critical points: P (u) =4u u u u Q x (u) = 336u u Q y (u) = 160u u Q t (u) =4u u u/ /2 ie, they are given by: P (u) =0, x = Q x(u) P 0 (u), y = Q y(u) P 0 (u), t = Q t(u) P 0 (u) Which one of these 4 is minimal? 18/25
23 Testing Minimality Def. F(z 1,,z n ) is combinatorial if every coefficient is 0. Prop. [PemantleWilson] In the combinatorial case, one of the minimal critical points has positive real coordinates. Thus, we add the equation for a new variable and select the positive real point(s) z with no from a new Kronecker representation: P (v) =0 P 0 (v)z 1 Q 1 (v) =0 P 0 (v)z n Q n (v) =0 P 0 (v)t Q t (v) =0.. This is done numerically, with enough precision. 19/25
24 Example F = 1 H = 1 (1 x y)(20 x 40y) 1 Critical point : x(2x + 41y 21) = y(41x + 80y 60) 4 critical points, 2 of which are real: (x 1,y 1 )=(0.2528, ), (x 2,y 2 )=( , ) Add H(tx, ty) =0and compute a Kronecker representation: P (u) =0, x = Q x(u) P 0 (u), y = Q y(u) P 0 (u), Solve numerically and keep the real positive sols: t = Q t(u) P 0 (u) (0.31, 0.55, 0.99), (0.31, 0.55, 1.71), (0.25, 9.99, 0.09), (0.25, 0.99, 0.99) (x 1,y 1 ) is not minimal, (x 2,y 2 ) is. 20/25
25 Algorithm and Complexity Thm. If F (z) is combinatorial, then under regularity conditions, the points contributing to dominant diagonal asymptotics can be determined in Õ(hd 5 D 4 ) bit operations. Each contribution has the form A k = T k k (1 n)/2 (2 ) (1 n)/2 (C + O(1/k)) T, C can be found to 2 apple precision in Õ(h(dD) 3 + Dapple) bit ops. This result covers the easiest cases. The complexity should be compared with the size of the result. All conditions hold generically and can be checked within the same complexity, except combinatoriality. [Melczer-S.16] 21/25
26 Example: Apéry's sequence 1 1 t(1 + x)(1 + y)(1 + z)(1 + y + z + yz + xyz) =1+ +5xyzt + Kronecker representation of the critical points: a = 2u 1006 P 0 (u) P (u) =u 2 366u u , b = c = P 0, z = (u) P 0 (u) There are two real critical points, and one is positive. After testing minimality, one has proven asymptotics 22/25
27 Example: Restricted Words in Factors F (x, y) = 1 x 3 y 6 + x 3 y 4 + x 2 y 4 + x 2 y 3 1 x y + x 2 y 3 x 3 y 3 x 4 y 4 x 3 y 6 + x 4 y 6 words over {0,1} without or /25
28 Minimal Critical Points in the Noncombinatorial Case Then we use even more variables and equations: H(z) =0 1 = = n H(u) =0 u 1 2 = t z 1 2,..., u n 2 = t z n 2 + critical point equations for the projection on the t axis And check that there is no solution with t in (0,1). Prop. Under regularity assumptions, this can be done in Õ(hd 4 2 3n D 9 ) bit operations. 24/25
29 Summary & Conclusion D-finite diag. alg. rat. Diagonals are a nice and important class of generating functions for which we now have many good algorithms. ACSV can be made effective (at least in simple cases). Requires nice semi-numerical Computer Algebra algorithms. Without computer algebra, these computations are difficult. Work in progress: extend beyond some of the assumptions (see Melczer s thesis). The End
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