Diagonals of rational functions

Transcription

1 Diagonals of rational functions Main Conference of Chaire J Morlet Artin approximation and infinite dimensional geometry 27 mars 2015 Pierre Lairez TU Berlin

2 Diagonals: definitions and properties Binomial sums Computing diagonals Diagonals: definitions and properties Binomial sums Computing diagonals

3 Diagonal of a power series Définition f = a i1,,i n x i1 1 x i n n Q x 1,,x n i 1,,i n N n diag f def = a i,,i t i i 0

4 A combinatorial problem Counting rook paths y (7, 10) 7 a i,j def = nb of rook paths from (0,0) to (i,j) Easy recurrence: a i,j = a k,j + k <i k <j a i,k (0,0) 2 1 x What about a n,n? asymptotic? existence of a recurrence?

5 Recurrence relations for rook paths dimension 2 9nu n ( n)u n+1 + (2 + n)u n+2 = 0 dimension 3 192n 2 (1 + n)( n)u n +(1 + n)( n n n 3 )u n+1 (2 + n)( n n n 3 )u n+2 +2(2 + n)(3 + n) 2 ( n)u n+3 = 0 dimension n 3 (1 + n) 2 ( n n n n 4 )u n (1 + n) 2 ( n n n 7 )u n+1 +2(2 + n)( n n n 8 )u n+2 (3 + n) 2 ( n n n 7 )u n+3 +2(3 + n) 2 (4 + n) 3 ( n n n n 4 )u n+4 = 0

6 Differential equation for diagonals a i,j = a k,j + a i,k a i,j x i y j = k <i k <j i,j 0 n 0 a n,n t n = diag 1 1 x 1 x y 1 y 1 1 x 1 x y 1 y Theorem (Lipshitz 1988) diagonal differentially finite If R Q(x 1,,x n ) Q x 1,,x n, then diag R satisfies a linear differential equation with polynomial coefficients c r (t)y (r ) + + c 1 (t)y + c 0 (t)y = 0

7 More properties of diagonals Theorem (Furstenberg 1967) algebraic diagonal If f (t) = a n t n is an algebraic series (ie 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

8 Algebricity modulo p Example f = n ( (3n)! n! 3 tn = diag 1 1 x y z ) is not algebraic f (1 + t) 1 4 mod 5 f ( 1 + 6t + 6t 2) 1 6 mod 7 f ( 1 + 6t + 2t 2 + 8t 3) 1 10 mod 11 Besides, ( 27t 2 t ) f + (54t 1)f + 6f = 0

9 Proof of algebricity modulo p F q, the base field Slicing operators For r Z, E r i a i t i def = i a qi+r t i and E r I a I x I def = a qi+(r,,r ) x I I We check 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,,xq n; If f (t) = i a i t i, then f (t) = 0 r <q t r i a qi+r t qi

10 Proof of algebricity modulo p F q, the base field Slicing operators For r Z, E r i We check a i t i def = diag E r = E r diag ; x i E r (F) = E r (x q i F) ; i a qi+r t i and E r I a I x I def = G(x)E r (F) = E r (G(x) q F), because G(x q ) = G(x) q, where x q = x q 1,,xq n; If f (t) = i a i t i, then a qi+(r,,r ) x I I f (t) = 0 r <q t r i a qi+r t i q

11 Proof of algebricity modulo p F q, the base field Slicing operators For r Z, E r i a i t i def = i a qi+r t i and E r I a I x I def = a qi+(r,,r ) x I I We check 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,,xq n; If f (t) = i a i t i, then f (t) = 0 r <q t r E r (f ) q

12 Proof of algebricity modulo p Let R = A F F q(x), d = max(deg A,deg F) and the F q -vector space V = { diag ( ) P deg P d } ( ( F F q t dim V d+n )) n 1 Operators E r stabilize V Proof ( ) P PF E r diag = diag E q 1 F r F q = diag E r (PF q 1 ) V F

13 Proof of algebricity modulo p Let R = A F F q(x), d = max(deg A,deg F) and the F q -vector space V = { diag ( ) P deg P d } ( ( F F q t dim V d+n )) 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 i, f i = c ij f q j Proof f i = 0 r <q ( = j t r E r (f i ) q = 0 r <q j b ij t r ) f q j 0 r <q t r ( j b ij f j ) q

14 Proof of algebricity modulo p Let R = A F F q(x), d = max(deg A,deg F) and the F q -vector space V = { diag ( ) P deg P d } ( ( F F q t dim V d+n )) 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 i, f i = c ij f q j j 3 All the elements of V are algebraic Proof i, f q i = j cq ij f q2 j, etc { Thus, over F q (t), Vect (R) } qk 0 k s } Vect {f qk i 0 k s, 1 i s Vect { f qs i 1 i s}

15 Characterization of diagonals? Conjecture (Christol 1990) integer coefficients + convergent + diff finite diagonal If a n t n Z t, has radius of convergence > 0, and satisfies a linear differential equation with polynomial coefficients, then it is the diagonal of a rational power series 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 ( ) N n (3n)! t n = 3 n! 3 Question : Find a f such that 3 < N(f ) <

16 Diagonals: definitions and properties Binomial sums Computing diagonals Diagonals: definitions and properties Binomial sums Computing diagonals

17 Binomial sums Examples 2n k=0 n ( ) 2 ( ) 2 n n + k = k k k=0 n i=0 j=0 ( ) 3 2n ( 1) k = ( 1) n (3n)! k (n!) 3 k=0 ( n k ) ( ) k n + k k j=0 n ( ) 2 ( ) i + j 4n 2i 2j = (2n + 1) i 2n 2i ( 1) ( ) ( n+r +s n n r s r 0 s 0 ) ( n+s s ) ( n+r r (Dixon) ( ) 3 k (Strehl) j ( ) 2 2n n ) ( 2n r s ) ( n = n 4 k) k 0

18 Binomial sums Further examples Number theory n 3 u n + (n 1) 3 u n 2 = (34n 3 51n n 5)u n 1 n ( ) 2 ( ) 2 n n + k avec u n = (Apéry) k k k=0 Important step in proving that ζ(3) = n 1 1 n 3 Analysis of algorithm Q [50] Develop computer programs for simplifying sums that involve binomial coefficients Exercise The Art of Computer Programming Knuth (1968)

19 Definition δ : Z Q is a binomial sum (δ 0 = 1 et δ n = 0 si n 0) n Z a n Q is a binomial sum for all a Q (n,k) Z 2 ( n k) Q is a binomial sum If u,v : Z p Q are bs, then u + v and uv are bs If u : Z p Q are bs and λ : Z p Z q is an affine map, then u λ is a bs If u : Z p Q is a bs, then n 1 (n 1,,n p ) Z p k=0 u k,n2,,n p Q is a bs

20 Diagonals binomial sums Theorem (Bosta, Lairez, Salvy 2014) A sequence (u n ) n 0 is a binomial sum if and only if its generating function u n t n is the diagonal of a rational series Example n n 0 k=0 ( n k ) 2 ( n+k k ) 2 ( t n = diag 1 (1 x 1 )(1 x 2 )(1 x 3 ) x 4 (x 1 +x 2 x 3 x 1 x 2 x 3 ) )

21 Corollaries Corollary 1 binomial sum recurrence If (u n ) n 0 is a binomial sum, then it satisfies a linear recurrence with polynomial coefficients Corollary 2 algebraic gf binomial sum If u n t n is an algebraic series then (u n ) n 0 is a binomial sum Corollary 3 binomial sum algebraic gf mod p If (u n ) n 0 is a binomial sum, then u n t n is an algebraic series modulo p for almost all prime p Conjecture integral + exp bounded + recurrence binomial sum If (u n ) n 0 Z N grows at most exponentially and satisfies a linear recurrence with polynomial coefficients, then it is a binomial sum

22 Binomial sums as diagonals Skectch of the proof Proposition All binomial sums are linear combinations of sequences in the form (k 1,,k p ) Z p [1] ( R 0 R k 1 1 R k ) d d, where R 0,,R p Q(x 1,,x d ) With one variable, [1] (RS n ) = [x1 n xn d+1 ]( R ) 1 x 1 x d+1 S } {{ } may not be a power series

23 Diagonals as binomial sums Skectch of the proof Let R = 1 + a 1 x m x m Q(x 1,,x d ) r ( ) R = x m k1 + + k 0 r a k 1 k k N r 1,,k 1 ak r r x k 1m 1 x k r m r r } {{ } ( k1 + +k r k 1,,k r x m 0 ) = ( k1 + +k r binomial sum [x n 1 xn d ]R = k 1 C k ) ( k2 + +k r k 2 k Z e C k 1 Γ (n,k) où ) ( kr ) 1 +k r k r, so Ck is a Γ = { e (n,k) R R e + m 0 + k i m i = (n,,n) } i=1

24 Diagonals: definitions and properties Binomial sums Computing diagonals Diagonals: definitions and properties Binomial sums Computing diagonals

25 Algorithmic Lipshitz theorem Theorem (Lipshitz 1988) diagonal differentially finite If R Q(x 1,,x n ) Q x 1,,x n, then diag R satisfies a linear differential equation with polynomial coefficients c r (t)y (r ) + + c 1 (t)y + c 0 (t)y = 0 How to compute the differential equation? Once computer, we get everything we want about the diagonal It would allow to prove identities between binomial sums

26 Diagonals as integrals Basic fact: C, diag ( x i C x i x j C x j ) = 0 Consider the following transformation T : Q(x 1,,x n ) Q(t,x 1,,x n 1 ) ( ) 1 t R R x 1,,x n 1, x 1 x n 1 x 1 x n 1 T (R) = n 1 i=1 diag R = 1 (2iπ) n C i diag R = 0 x i T (R)dx 1 dx n 1

27 Computing integrals K a field of characteristic 0 with a derivation δ (usually K = Q(t) and δ = t ) R = a f K(x 1,,x n ) Problem Find c 0,,c r K such that C 1,,C n K(x) c r δ r (R) + + c 1 δ(r) + c 0 (R) = n i=1 C i x i Problem (bis) Compute a basis and normal forms in K(x 1,,x n )/ n i=1 x i K(x 1,,x n )

28 Computing integrals K a field of characteristic 0 with a derivation δ (usually K = Q(t) and δ = t ) R = a f K(x 1,,x n ) Problem Find c 0,,c r K such that C 1,,C n K(x) c r δ r (R) + + c 1 δ(r) + c 0 (R) = n i=1 C i x i Problem (bis) Compute a basis and normal forms in K[x 1,,x n, f 1 ]/ =: H n Rham (An K n i=1 x i K[x 1,,x n, f 1 ] \ V(f )) Hn 1 Rham (V(f ))

29 Finiteness of the de Rham cohomology Theorem (Grothendieck 1966) H n Rham (An K \ V(f )) is a finite dimensional K-vector space Theorem (Griffiths 1969) dim K H n Rham (An K \ V(f )) < (deg f + 1)n Corollary The diagonal of R(x 0,,x n ) is solution of a linear differential equation with polynomial coefficients of order at most (d + 1) n, where d is the degree of the denominator of T (R)

30 Reduction of pole order Homogeneous case f K[x 0,,x n ] homogeneous def V f = { a f homogeneous of degree n 1 } q def Notation : i = x i c Fact : i f q 1 = ic f q 1 (q 1)c i f f q Rewriting rule i c i i f f q 1 q 1 i i c i f q 1 (maps V f V f ) Theorem (Griffiths 1969) If V(f ) P n K is smooth then a f q V a f, f q = c i i f s a f q 0 i

31 Reduction of pole order Homogeneous singular case Rewriting rules are ambiguous: if i c i i f = 0, then 0 i i c i f q We can add the rules i i c i 0, it still preserves equivalence classes modulo derivatives New reductions 0 rk r b f appear, we add the rules b rk r + 1 q f 0 q f q rg 2 Theorem There exists an r > 0 such that for all a f q V f a f q V f, a f q = i i c i f s a f q rk r 0 Leads to an efficient algorithm for computing rational integrals (Lairez 2015)

32 An example f = 2xyz(w x)(w y)(w z) w 3 (w 3 w 2 z + xyz) e(q,r) : number of indepedent rational functions a/f q that are not reducible with rules of rank r q q > 4 no rule q 3 e(q, 1) q e(q, 2) e(q, 3) dim H 3 (P 3 \ V(f )) = 10