Fast computation
of the N th term of an algebraic series
in positive characteristic
Alin Bostan
1
Gilles Christol
1 Inria
2 Institut
2
Philippe Dumas
1
Saclay
de mathématiques de Jussieu
Functional Equations
in
Limoges, March 29th 2016
1 / 30
Motivation
I
I
Ubiquity of algebraic functions (combinatorics, number
theory, algebraic geometry)
Confluence of several domains:
– functional equations
– automatic sequences
– complexity theory
I
One of the most difficult questions in modular
computations is the complexity of computations
mod p for a large prime p of coefficients in the
expansion of an algebraic function.
D. Chudnovsky & G. Chudnovsky, 1990
Computer Algebra in the Service of
Mathematical Physics and Number Theory
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Problem and main result
Input:
– field K = Fp
– E(x, y) ∈ K[x, y], with E(0, 0) = 0, Ey (0, 0) 6= 0
– N ∈ N≥0
Output:
– the N th coefficient of the unique solution f (x) ∈ K[[x]] of
E(x, f (x)) = 0, f (0) = 0
Main result:
– arithmetic complexity linear in log N and almost linear in p
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State of the art
First N coefficients
Method
char. 0
char. p
Undetermined coefficients
Fixed point iteration
Newton iteration
Linear recurrence
O(N d )†
X
X¶
X¶
X∗
† d = deg
y
E(x, y)
¶ FFT
used all along
e 2 )¶
O(N
e )¶
O(N
O(N )
∗ with
Kung + Traub, 1976
All algebraic functions can be
computed fast
p-adic computations
Chudnovsky + Chudnovsky, 1986
On expansion of algebraic functions in power and Puiseux series, I
4 / 30
State of the art
Only N th term (with precomputation)
char. 0
char. p
O(log2 N ) + O(1)
√
e N ) + O(1)
O(
X
Method
Binary powering (d = 1 only)
Baby steps – Giant steps
Divide and Conquer
X∗
e 3d )
O(logp N ) × O(p
Rational series by binary powering
Miller + Brown, 1966,
An algorithm for
evaluation of remote terms in
a linear recurrence sequence
Fiduccia, 1985
An efficient formula for
linear recurrences
5 / 30
State of the art
Only N th term (with precomputation)
Method
Rational series
Baby steps – Giant steps
Divide and Conquer
char. 0
char. p
O(log2 N ) + O(1)
√
e N ) + O(1)
O(
X
X∗
e 3d )
O(logp N ) × O(p
Baby steps – Giant steps after precomputing
algebraic equation −→ differential equation −→ linear recurrence
Chudnovsky + Chudnovsky, 1988
Approximations and
complex multiplication
according to Ramanujan
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State of the art
Only N th term (with precomputation)
Method
Rational series
Baby steps – Giant steps
Divide and Conquer
char. 0
char. p
O(log2 N ) + O(1)
√
e N ) + O(1)
O(
X
X∗
e 3d )
O(logp N ) × O(p
Divide and Conquer after precomputing
algebraic equation −→ Mahler equation −→ DAC recurrence
Christol + Kamae
+ Mendès France + Rauzy, 1980,
Suites algébriques, automates et
substitutions
7 / 30
From algebraic equation to Mahler equation
Algebraic equation y = 2x + 5xy + 4xy 2 + xy 3
y
1
y
y

0



 1



2
0
y3
y9
y 27
p=3
1
1+2x2 +x3
x3
1+2x2 +x3 +2x5 +x6 +x8 +x9 +2x11 +x12
x12
1+x
x
1+x+2x2 +x3 +x4
x4
1+x+2x2 +x3 +x4 +2x5 +2x6 +2x7 +x8 +x9 +x10 +2x11 +x12 +x13
x13
2
2 +x3
2 1+x+2x
3
2+2x+x2 +2x3 +2x4 +x5 +x6 +x7 +2x8 +2x9 +2x10 +x11 +2x12
x12
x








Mahler equation
2x2 + 2x3 + x4 f (x)
+ 1 + x2 + 2x3 + 2x4 + x5 + 2x6 f (x3 )
+ 2 + 2x3 + 2x5 + x6 + 2x9 f (x9 ) + x9 f (x27 ) = 0
Frobenius f (x)p = f (xp )
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From algebraic equation to Mahler equation
Algebraic equation y = 2x + 5xy + 4xy 2 + xy 3
Mahler equation
2x2 + 2x3 + x4 f (x)
+ 1 + x2 + 2x3 + 2x4 + x5 + 2x6 f (x3 )
+ 2 + 2x3 + 2x5 + x6 + 2x9 f (x9 ) + x9 f (x27 ) = 0
Divide-and-conquer recurrence
2fn−2 + 2fn−3 + fn−4
+ f n3 + f n−2 + 2f n−3 + 2f n−4 + f n−5 + 2f n−6
3
3
3
3
3
+ 2f n9 + 2f n−3 + 2f n−5 + f n−6 + 2f n−9 + f n−9 = 0
9
9
9
9
27
fx = 0 if x 6∈ N≥0
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+5x131 +2x130 +4x129 +x128 +2x127 +5x126 +3x125 +2x124 +4x123 +x122 +x121 +6x120 +6x119
From
algebraic
equation
to Mahler
equation
118
117
116
115
114
113
112
111
110
109
108
107
+2x
+2x
+5x
+3x
+6x
+4x
+2x
+6x
+4x
+6x
+5x
+6x
+5x106 +6x105 +4x104 +3x103 +4x102 +x101 +2x100 +5x99 +3x98 +4x97 +5x71 +4x69 +2x68 +2x67
+4x66 +5x65 +3x64 +x62 +2x57 +3x55 +3x54 +3x53 +6x52 +4x51 +5x50 +4x48 +2x46 +4x45
+3x44 +x43 +6x42 +5x41 +2x40 +5x39 +3x38 +6x37 +3x36 +2x35 +x34 +4x33 +3x32 +6x31 +5x30
+3x29 +4x28 +x27 +3x26 +6x25 +5x24 +5x23 +5x22 +2x21 +4x20 +x18 +2x17 +5x16 +5x15 +3x14
+4x13 +6x12 +2x11 +4x10 +2x9 +x8 +2x7 +5x6 +5x4 +3x3 +4x2 +1 f (x7 )
)
+ 6x165 +5x164 +2x163 +2x162 +4x161 +3x160 +2x155 +3x153 +2x151 +4x150 +3x149 +6x147
(
+x144 +2x143 +5x142 +4x141 +3x140 +6x139 +x137 +2x136 +5x135 +x134 +3x133 +5x132 +2x130
+4x129 +3x128 +4x127 +6x126 +6x125 +6x123 +5x122 +2x121 +2x120 +4x119 +3x118 +5x116
+3x115 +4x114 +4x113 +x112 +6x111 +4x106 +6x104 +4x102 +x101 +6x100 +5x98 +2x71 +3x69
+x67 +2x66 +5x65 +5x64 +3x63 +4x62 +5x57 +4x55 +5x53 +3x52 +4x51 +x49 +5x46 +3x45 +4x44
+6x43 +x42 +2x41 +5x39 +3x38 +4x37 +5x36 +x35 +4x34 +3x32 +6x31 +x30 +6x29 +2x28 +2x27
+2x25 +4x24 +3x23 +6x21 +6x18 +5x17 +2x16 +2x15 +4x14 +3x13 +2x8 +3x6 +2x4 +4x3 +3x2 +6 f (x49 )
)
+ x165 +2x164 +5x163 +5x162 +3x161 +4x160 +5x155 +4x153 +5x151 +3x150 +4x149 +x147 f (x343 )=0
(
)
O(logp N ) × pO(d)
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Sections
Section operators
X
X
Sr
fn xn =
fpk+r xk ,
n≥0
0≤r<p
k≥0
Lemma
Let f be in K[[x]] and let N = (N` · · · N1 N0 )p be the radix p
expansion of N . Then fN = (SN` · · · SN1 SN0 f )(0).
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Sections
N = 100000
p=7
= 5 × 16807 + 6 × 2401 + 4 × 343 + 3 × 49 + 5 × 7 + 5 × 1
= (5, 6, 4, 3, 5, 5)7
f (x) = f0 + f1 x + f2 x2 + f3 x3 + f4 x4 + f5 x5 + · · ·
S5 f (x) = f5 + f12 x + f19 x2 + f26 x3 + f33 x4 + f40 x5 + · · ·
S5 S5 f (x) = f40 + f89 x + f138 x2 + f187 x3 + f236 x4 + f285 x5 + · · ·
S3 S5 S5 f (x) = f187 + f530 x + f873 x2 + f1216 x3 + f1559 x4 + f1902 x5 + · · ·
S4 S3 S5 S5 f (x) = f1559 + f3960 x + f6361 x2 + f8762 x3 + f11163 x4 + · · ·
S6 S4 S3 S5 S5 f (x) = f15965 + f32772 x + f49579 x2 + f66386 x3 + f83193 x4 + · · ·
S5 S6 S4 S3 S5 S5 f (x) = f100000 + f217649 x + f335298 x2 + f452947 x3 + · · ·
S5 S6 S4 S3 S5 S5 f (0) = f100000
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Diagonal
F (x, y) =
y
y(1 − 5xy − xy 2 − 3xy 3 )
1 − 2x − 5xy − 4xy 2 − xy 3
+ 2xy
+ 4x2 y
+ x4 y 2
+ 3x3 y 3
+ 3xy 3
+ 5xy
p=7
+ 4x5 y
+ 2x4 y
+ x3 y
+ 3x2 y 2 + 5x3 y 2
4
=
2 4
+ 6x5 y 3 + 4x6 y 3
4 4
+ 6x y
5 4
+x y
+ 2x2 y 5 + 2x3 y 5
+ 5x2 y 7 + 6x3 y 7
+x y
+ 2x y
+x y
4 9
+x y
+ x9 y
+ 2x10 y
+ 6x9 y 2 + 3x10 y 2
+ 6x8 y 3
7 4
8 4
+ 3x y
+ 6x10 y 3
9 4
+ 5x y
+ 4x9 y 5 + 4x10 y 5
+ 5x6 y 6 + 6x7 y 6
+ 4x9 y 6 + 6x10 y 6
5 8
+ 3x y
5 9
+ 3x y
+ 6x7 y 7
6 8
+ 2x y
6 9
+ 6x y
+ 3x9 y 7 + 5x10 y 7
8 8
+ 3x y
+ 6x9 y 8 + 5x10 y 8
+ x9 y 9 + 6x10 y 9
7 9
+ 6x y
+ 6x5 y 10 + 2x6 y 10
+ 5x3 y 10
+ 4x8 y
+ x7 y 3
+x y
+ 5x5 y 7
4 8
+ 2x7 y
+ 6x5 y 5 + 4x6 y 5 + 5x7 y 5
+ 2x2 y 6 + 3x3 y 6 + 5x4 y 6
3 9
+ x6 y
+ 5x5 y 2 + 2x6 y 2 + 2x7 y 2
+ x8 y 10
+ 4x10 y 10
+ x4 y 11 + x5 y 11 + 5x6 y 11 + 4x7 y 11 + 4x8 y 11 + 2x9 y 11
+ 6x5 y 12 + 2x6 y 12 + x7 y 12
4 13
6 13
+ 5x y
+ 3x9 y 13 + x10 y 13
+ 5x y
5 14
7 14
+ 5x y
+ 3x9 y 12+ 3x10 y 12
+ 3x y
+ 4x9 y 14+ 2x10 y 14
+ 6x5 y 15 + x6 y 15 + 3x7 y 15 + x8 y 15 + 2x9 y 15+ 4x10 y 15
+ 5x5 y 16 + 4x6 y 16 + 4x7 y 16
+ 5x9 y 16+ 4x10 y 16
· ·3x
· 8 + x9 + 4x10 + x14 + 5x15 + · · ·
f (x) = DF (x) = 2x + 3x + 3x + x + 6x + 5x + 6x +
2
3
4
5
6
7
+
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Diagonal
Theorem (Furstenberg’s theorem)
Every algebraic series is the diagonal of a bivariate rational function.
E(x, f (x)) = 0,
f (x) = D
a
b
f (0) = 0,
Ey (0, 0) 6= 0
with a(x, y) = yEy (xy, y),
b(x, y) = E(xy, y)/y
Furstenberg, 1967,
Algebraic functions over
finite fields
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Actions
univariate sections Sr : action on formal series
a
Sr f = Sr D
b
diagonal and commutation rule
Sr D = DSr
?
bivariate sections Sr : action on bivariate rational functions
a
a
Sr D = DSr
b
b
Frobenius
?
pseudo-sections Tr : action on bivariate polynomials
a
Sr abp−1
Tr a
Christol, 1979,
DSr = D
=D
b
b
b
Ensembles
presque périodiques
k-reconnaissables
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Actions
univariate sections Sr : action on formal series
fN = (SN` · · · SN1 SN0 f )(0)
?
bivariate sections Sr : action on bivariate rational functions
?
pseudo-sections Tr : action on bivariate polynomials
(TN` · · · TN1 TN0 a)(0, 0)
fN =
b(0, 0)
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Finite dimensional stable subspace
DSr
a
Sr abp−1
Tr a
=D
=D
b
b
b
dx = max(degx a, degx b),
dy = max(degy a, degy b)
Fp [x, y]dx ,dy −→ Fp [x, y]pdx ,pdy −→
v 7−→
p−1
vb
Fp [x, y]dx ,dy
= vB 7−→ Sr vB = Tr v
Fp [x, y]dx ,dy left stable by Tr , 0 ≤ r < p
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Finite dimensional stable subspace
Fp [x, y]dx ,dy −→ Fp [x, y]pdx ,pdy −→
v 7−→
Fp [x, y]dx ,dy
vbp−1 = vB 7−→ Sr vB = Tr v
p = 11
E = (1+x)(x−y)+x2 y 2 +
(1 + x)y 3 + y 4
a = −y −xy 2 +3y 3 +4y 4 +
3xy 4 + 2x2 y 4
b = −1 + x − xy + y 2 +
x2 y + y 3 + xy 3 + x2 y 3
dx = 2, dy = 4
small box [0, dx ] × [0, dy ]
b
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Finite dimensional stable subspace
Fp [x, y]dx ,dy −→ Fp [x, y]pdx ,pdy −→
v 7−→
Fp [x, y]dx ,dy
vbp−1 = vB 7−→ Sr vB = Tr v
homothety with ratio p − 1
almost large box [0, (p − 1)dx ] ×
[0, (p − 1)dy ]
B = bp−1
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Finite dimensional stable subspace
Fp [x, y]dx ,dy −→ Fp [x, y]pdx ,pdy −→
v 7−→
Fp [x, y]dx ,dy
vbp−1 = vB 7−→ Sr vB = Tr v
v = xy 3 in the canonical basis
translation by (1, 3)
large box [0, pdx ] × [0, pdy ] xy 3 B
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Finite dimensional stable subspace
Fp [x, y]dx ,dy −→ Fp [x, y]pdx ,pdy −→
v 7−→
Fp [x, y]dx ,dy
vbp−1 = vB 7−→ Sr vB = Tr v
filter r = 3
xy 3 B
21 / 30
Finite dimensional stable subspace
Fp [x, y]dx ,dy −→ Fp [x, y]pdx ,pdy −→
v 7−→
Fp [x, y]dx ,dy
vbp−1 = vB 7−→ Sr vB = Tr v
contraction
small box [0, dx ] × [0, dy ]
T3 xy 3 = S3 xy 3 B
22 / 30
Precomputation of A0 , A1 , . . . , Ap−1
All information is in B = bp−1 .
matrix Ar : xn y m −−−−−−−−−→ xn y m B −−−−−−−−−→ Sr xn y m B
translation
selection
No computation in K = Fp , except raising b to the power p − 1
e 2 ) (binary powering, Kronecker substitution, FFT)
Cost: O(p
Theorem
The N th coefficient can be computed in time
e 2 ) + O(logp N ).
O(p
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Only a small part of B = bp−1 is enough
Task: Computation of A0 , A1 , . . . , Ap−1
row index i = (k, `)
column index j = (n, m)
B=
X
cα,β xα y β
α,β
xn y m −−−−−−−−−→ xn y m B −−−−−−−−−→ Sr xn y m B
translation
selection
X
X
cα,β xn+α y m+β −−−−→
cα,β xk y `
xn y m −−−−→
α,β
(C)
n + α = pk + r
m + β = p` + r
α,β
(C)
=⇒ β − α = p(` − k) + n − m
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Only a small part of B = bp−1 is enough
p = 11
p = 109
25 / 30
Improved precomputation
B(x/t, t) =
X
cα,β xα tβ−α =
α,β
X
πδ (x)tδ
δ
B(x/t, t) = b(x/t, t)p−1 =
b(x/t, t)p
b(xp /tp , tp )
=
b(x/t, t)
b(x/t, t)
X
1
=
cu (x)tu
b(x/t, t)
u
b(xp /tp , tp ) =
rational series
πδ (x) =
X
P
v bv (x
p )tpv
for free
cu (x)bv (xp )
u+pv=δ
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Improved precomputation
δ = β − α = intercept of the strips with
the ordinates axis
e
big leaps from one strip to the next O(p)
(Kronecker substitution, FFT, Newton iteration)
Fiduccia, 1985
An efficient formula for
linear recurrences
27 / 30
Main result
Theorem
Let E be in Fp [x, y]h,d satisfy E(0, 0) = 0 and Ey (0, 0) 6= 0, and
let f ∈ Fp [[x]] be its unique root with f (0) = 0. One can
compute the coefficient fN of f in
e
h2 (d + h)2 logp N + O(h(d
+ h)5 p)
operations in Fp .
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Timings
Maple implementation; timings on Intel Core i5, 2.8 GHz, 3MB.
k
With p = 9001, N = 1010 , k = 1..6,
ComputingTime(N ) ' 0.0011 · logp (N ) − 1.3563
With p = NextPrime(2k ), k = 1..12,
PrecomputingTime(p) ' 0.00037 · p log(p) + 0.04835
Documents at url
http://specfun.inria.fr/dumas/Research/AlgModp/
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Theorem
Let E be in Fp [x, y]h,d satisfy E(0, 0) = 0 and Ey (0, 0) 6= 0, and
let f ∈ Fp [[x]] be its unique root with f (0) = 0. One can
compute the coefficient fN of f in
e
h2 (d + h)2 logp N + O(h(d
+ h)5 p)
operations in Fp .
Thanks for your attention!
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