Congruences modulo cyclotomic polynomials and algebraic independence for $q$-series

0.19 MB·

by  Boris Adamczewski

CONGRUENCES MODULO CYCLOTOMIC POLYNOMIALS AND
ALGEBRAIC INDEPENDENCE FOR q-SERIES
B. ADAMCZEWSKI, J. P. BELL, É. DELAYGUE AND F. JOUHET
7
1
0 Abstract. Weprovecongruencerelationsmodulocyclotomicpolynomialsformultisums
2
of q-factorial ratios, therefore generalizing many well-known p-Lucas congruences. Such

n congruences connect various classical generating series to their q-analogs. Using this, we
a
proveapropagationphenomenon: whenthesegeneratingseriesarealgebraicallyindepen-
J
dent, this is also the case for their q-analogs.
3
2
1. Introduction and main results
]
O
After the seminal work of Lucas [10], a great attention has been paid on congruences
C
modulo prime numbers p satisfied by various combinatorial sequences related to binomial
.
h
coefficients. A typical example of these so-called p-Lucas congruences is given by:
t
a
r r r
m 2(m+np) 2m 2n
mod p, (1.1)
[ m+np ≡ m n
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)

1 where0 m p 1,r 1,andn 0. Intermsofgeneratingseriesthesecongruences(1.1)
v ≤ ≤ − ≥ ≥
translate as
8
7 g (x) A(x)g (xp) mod pZ[[x]], (1.2)
3 r ≡ r
6 where
0 ∞ r
2n
1. gr(x) := xn
n
0
n=0(cid:18) (cid:19)
7 X
1 and A(x) is a polynomial (depending on r and p) in Z[x] of degree at most p 1. This
−
: functional point of view led the authors of [2] to define general sets of multivariate power
v
i series including the following one which is of particular interest for our purpose.
X
r Definition 1.1. Let d be a positive integer and x = (x ,...,x ) be a vector of indetermi-
a 1 d
nates. We let L denote the set of all power series g(x) in Z[[x]] with constant term equal
d
to 1 and such that for every prime number p:
(i) there exist a positive integer k and a polynomial A in Z[x] satisfying
g(x) A(x)g xpk mod pZ[[x]].
≡
(ii) deg (A) pk 1 for all i, 1 i (cid:0)d. (cid:1)
xi ≤ − ≤ ≤
This project has received funding from the European Research Council (ERC) under the European
Union’s Horizon 2020 researchand innovation programme under the Grant Agreement No 648132.
1
Using p-adic computations inspired by works of Christol and Dwork, it was proved in [2]
that a large family of multivariate generalized hypergeometric series belongs to L . This
d
provides, by specialization, a unified way to reprove most of known p-Lucas congruences as
well as to find many new ones. In addition, a general method to prove algebraic indepen-
dence of power series whose coefficients satisfy p-Lucas type congruences was developed.
Let us illustrate this approach with the following example. In 1980, Stanley [14] conjec-
tured that the series g are transcendental over C(x) except for r = 1, in which case we
r
have g (x) = (√1 4x)−1. He also proved the transcendence for r even. The conjecture
1
−
was solved independently by Flajolet [6] through asymptotic considerations and by Sharif
and Woodcock [13] by using the previously mentioned Lucas congruences. Incidentally,
this result is also a consequence of the interlacing criterion proved by Beukers and Heck-
man [3] for generalized hypergeometric series. Though there are three different ways to
obtain the transcendence of g for r 2, not much was apparently known about their
r
≥
algebraic independence, until Congruence (1.1) was used in [1, 2] to prove the following
result: all elements of the set g (x) : r 2 are algebraically independent over C(x).
r
{ ≥ }
In the present work, we aim at generalizing the approach of [2] in the setting of q-series.
It started with the following observation which can be derived from [7, 12, 15]:
r r r
2(m+nb) 2m 2n
mod φ (q)Z[q], (1.3)
b
m+nb ≡ m n
(cid:20) (cid:21)q (cid:20) (cid:21)q(cid:18) (cid:19)
where n,m,b,r are nonnegative integers with b 1, 0 m b 1, and φ (x) denotes
b
≥ ≤ ≤ −
the bth cyclotomic polynomial over Q. Here, for every complex number q, the central
q-binomial coefficients are defined as
2n [2n] ! n 1 qi
:= q N[q], where [n] ! := −
n [n] !2 ∈ q 1 q
(cid:20) (cid:21)q q i=1 −
Y
is the q-analog of n!. It is implicitly considered as a polynomial in q so that the formula is
still valid for q = 1. In particular, one has [n] ! = n! and the congruence (1.3) allows one
1
to recover (1.1) since φ (1) = p. Again in terms of generating series, (1.3) translates as
p
f (q;x) A(q;x)g (xb) mod φ (q)Z[q][[x]], (1.4)
r r b
≡
where A(q;x) is a polynomial in Z[q][x] of degree (in x) at most b 1 and we have set
−
∞ r
2n
f (q;x) := xn.
r
n
n=0(cid:20) (cid:21)q
X
This provides anarithmetic connection between the generating series g (x) anditsq-analog
r
f (q;x). It leads us to associate a set (q;g) of q-deformations with every element g in
r
D
L . We stress that (q;g) is closed under q-derivation.
d
D
Definition 1.2. Let q be a fixed nonzero complex number. Let g(x) be a power series in
L . We let (q;g) denote the set of all nonzero power series f(q;x) in Z[q][[x]] such that
d
for all integeDrs b 1 there exists a polynomial A(q;x) with coefficients in Z[q] satisfying:
≥
f(q;x) A(q;x)g xb mod φ (q)Z[q][[x]].
b
≡
2
(cid:0) (cid:1)
Our first result shows a propagation phenomenon of algebraic independence from gen-
erating series in L to their q-analogs.
d
Theorem 1.3. Let q be a nonzero complex number. Let g (x),...,g (x) be power series
1 n
in L , which are algebraically independent over C(x). Then for any f (q;x) in (q;g ),
d i i
1 i n, the series f (q;x),...,f (q;x) are also algebraically independent over CD(x).
1 n
≤ ≤
It immediately implies that all elements of the set f (q;x) : r 2 are algebraically
r
{ ≥ }
independent over C(x) for all nonzero complex numbers q. More generally, there is a
long tradition for combinatorists in studying q-analogs of famous sequences of natural
numbers, as the additional variable q gives the opportunity to refine the enumeration of
combinatorialobjectscountedbytheq = 1case. Tosomeextent, thenatureofagenerating
series reflects the underlying structure of the objects it counts [4]. By nature, we mean
for instance whether the generating series is rational, algebraic, or D-finite. In the same
line, algebraic (in)dependence of generating series can be considered as a reasonable way
to measure how distinct families of combinatorial objects may be (un)related.
It is known from [2] that many generating series g of multisums of factorial ratios belong
to L . For such series g, we will define q-analogs and prove that they lie in the set (q;g).
d
D
This will yield at once algebraic independence results, but also many generalizations of
Congruence (1.3). Finding congruences with respect to cyclotomic polynomials is actually
not a recent problem (see for instance [12, 8, 11] and the references cited there).
Our second result below is a general congruence relation extending (1.3) to the mul-
tidimensional case, by considering q-factorial ratios in the spirit of the ones in [16]. For
positive integers d,u,v, let e = (e ,...,e ) and f = (f ,...,f ) be tuples of vectors in Nd.
1 u 1 v
For n Nd, we define a q-analog of multidimensional factorial ratios (see Section 2 for
∈
precise notations) by:
[e n] ! [e n] !
(q;n) := 1 · q ··· u · q
Qe,f [f n] ! [f n] ! ·
1 q v q
· ··· ·
Furthermore, we consider the Landau step function ∆ defined on Rd by
e,f
u v
∆ (x) := e x f x .
e,f i j
⌊ · ⌋− ⌊ · ⌋
i=1 j=1
X X
We also define e = u e , f = v f , and set:
| | i=1 i | | j=1 j
e,f :=Px [0,1)d :Pthere is t in e or f such that t x 1 .
D ∈ · ≥
Proposition 1.4. Le(cid:8)t e and f be two tuples of vectors in Nd such tha(cid:9)t e = f and
∆ is greater than or equal to 1 on . Then, for every positive integer|b|, eve|ry|a in
e,f e,f
0,...,b 1 d and every n in Nd, weDhave (q;n) Z[q] and
e,f
{ − } Q ∈
(q;a+nb) (q;a) (1;n) mod φ (q)Z[q].
e,f e,f e,f b
Q ≡ Q Q
Proposition1.4extendsmanyknownresults,bothforq-analogsandp-Lucascongruences.
For instance, choosing d = 1, u = 1, v = 2, e = 2, and f = f = 1 yields (1.3), while
1 1 2
taking b prime and q = 1 allows one to recover Proposition 8.7 in [2]. As we will see in
3
Section3,Proposition1.4alsoleadstocongruencesfor(multi-)sumsofq-factorialratios. As
an illustration, we provide below two examples connected to the famous Apéry sequences.
Proposition 1.5. Consider for a given nonnegative integer t the following q-analogs of
the Apéry sequences
n 2 n 2 2
n n+k n n+k
a (q) := qtk and b (q) := qtk .
n n
k k k k
k=0 (cid:20) (cid:21)q(cid:20) (cid:21)q k=0 (cid:20) (cid:21)q(cid:20) (cid:21)q
X X
Then, for all nonnegative integers n,m,b with b 1, 0 m b 1, we have
≥ ≤ ≤ −
a (q) a (q)a (1) mod φ (q)Z[q] and b (q) b (q)b (1) mod φ (q)Z[q].
m+nb m n b m+nb m n b
≡ ≡
Setting
F (q;x) := (q;n)xn
e,f e,f
Q
n∈Nd
X
and assuming the conditions of Proposition 1.4, we obtain that F (q;x) belongs to
e,f
(q;F (1;x)), as F (1;x) is in L by [2, Proposition 8.7]. Theorem 1.3 therefore implies
e,f e,f d
Dthat algebraic independence among series F (1;x) propagates to their corresponding q-
e,f
analogs F (q;x). As noticed before, this holds for the series g (x) and their q-analogs
e,f r
f (q;x). Proposition 1.4 and a result about specializations of the series F (q;x) (stated
r e,f
in Section 3) actually provide much more general results, such as the following one.
Proposition 1.6. For a fixed nonzero complex number q, let be the set formed by the
q
F
union of the three following sets:
∞ n r ∞ n r r
n n n+k
xn : r 3 , xn : r 2
k ≥ k k ≥
(n=0 k=0(cid:20) (cid:21)q ) (n=0 k=0(cid:20) (cid:21)q(cid:20) (cid:21)q )
XX XX
and
∞ n 2r r
n n+k
xn : r 1 .
k k ≥
(n=0 k=0(cid:20) (cid:21)q (cid:20) (cid:21)q )
XX
Then all elements of are algebraically independent over C(x).
q
F
Proposition 1.6 is derived from Proposition 1.2 in [2], which corresponds to the case
q = 1.
In the next section, we fix some notation and recall basic facts about Dedekind domains.
In Section 3, we focus on congruence relations modulo cyclotomic polynomials and prove
Proposition 1.4. We also show how to derive results like Propositions 1.5 and 1.6. Finally,
the last section is devoted to a sketch of proof of Theorem 1.3.
2. Background and notations
Letusintroducesomenotationandbasicfactsthatwillbeusedthroughoutthisextended
abstract. Let dbeapositive integer. Givend-tuplesof realnumbers m = (m ,...,m ) and
1 d
n = (n ,...,n ), we set m+n := (m +n ,...,m +n ) and m n := m n + +m n .
1 d 1 1 d d 1 1 d d
· ···
4
If moreover λ is a real number, then we set λm := (λm ,...,λm ). We write m n if we
1 d
have m n for all k in 1,...,d . We also set 0 := (0,...,0) and 1 := (1,...,≥1).
k k
Polyno≥mials. Given a {d-tuple o}f natural numbers n = (n ,...,n ) and a vector of
1 d
indeterminates x = (x ,...,x ), we will denote by xn the monomial xn1 xnd. The
1 d 1 ··· d
(total) degree of such a monomial is the nonnegative integer n + + n . Given a ring
1 d
R and a polynomial P in R[x], we denote by degP the (total) de·g·r·ee of P, that is the
maximum of the total degrees of the monomials appearing in P with nonzero coefficient.
The partial degree of P with respect to the indeterminate x is denoted by deg (P).
i xi
Algebraic power series and algebraic independence. Let K be a field. We denote by
K[[x]] the ring of formal power series with coefficients in K and associated with the vector
of indeterminates x. We say that a power series f(x) K[[x]] is algebraic if it is algebraic
over the field of rational functions K(x), that is, if th∈ere exist polynomials A ,...,A in
0 m
K[x],notallzero,suchthatA (x)+A (x)f(x)+ +A (x)f(x)m = 0. Otherwise, f issaid
0 1 m
to be transcendental. Let f ,...,f be in K[[x]·].··We say that f ,...,f are algebraically
1 n 1 n
dependent if they are algebraically dependent over the field K(x), that is, if there exists a
nonzero polynomial P(Y ,...,Y ) in K[x][Y ,...,Y ] such that P(f ,...,f ) = 0. When
1 n 1 n 1 n
there is no algebraic relation between them, the power series f ,...,f are said to be
1 n
algebraically independent (over K(x)).
Dedekind domains. Let R be a Dedekind domain; that is, R is Noetherian, integrally
closed, and every nonzero prime ideal of R is a maximal ideal. Furthermore, any nonzero
element of R belongs to at most a finite number of maximal ideals of R. In other words,
given an infinite set of maximal ideals of R, then one always has p = 0 . For
S p∈S { }
every power series f(x) = a(n)xn with coefficients in R, we set
n∈Nd T
f (Px) := a(n) mod p xn (R/p)[[x]].
|p
∈
n∈Nd
X (cid:0) (cid:1)
The power series f is called the reduction of f modulo p. Let K denote the field of
|p
fractions of R. The localization of R at a maximal ideal p is denoted by R . Recall here
p
that R can be seen as the following subset of K:
p
R = a/b : a R,b R p .
p
{ ∈ ∈ \ }
Then R is a discrete valuation ring and the residue field R /p is equal to R/p.
p p
3. Some general congruences and appplications
We first give the detailed proof of Proposition 1.4, and we will then see how to derive
results like Propositions 1.5 and 1.6.
Proof of Proposition 1.4. In this proof, we write for , ∆ for ∆ and for .
e,f e,f e,f
Q Q D D
Recall that for all nonnegative integers n we have
1 qn n
− = φ (q) [n] ! = φ (q)⌊n/b⌋,
b q b
1 q ⇒
− b|n,b≥2 b=2
Y Y
5
from which we deduce, by definition of the step function ∆,
∞
(q;n) = φ (q)∆(n/b). (3.1)
b
Q
b=2
Y
As e = f , the function ∆ is 1-periodic in each of its variable and one easily obtains
from| |(3.1)| t|hat (q;n) is in Z[q] for every n in Nd if, and only if ∆ is nonnegative over
Q
Rd. This proves the first part of our proposition.
Let x be a complex variable. As e = f , we derive
| | | |
u ei·a(1 xei·nb+k)
(x;a+nb) = (x;nb) i=1 k=1 − .
Q Q Qvj=1Qkfj=·a1(1−xfj·nb+k)
If a/b is not in , then for each t in e or f,Qno elQement of 1,...,t a is divisible by b.
D { · }
Hence, if ξ is a complex primitive bth root of unity, then we have
b
u ei·a(1 ξk)
(ξ ;a+nb) = (ξ ;nb) i=1 k=1 − b = (ξ ;nb) (ξ ;a),
Q b Q b v fj·a(1 ξk) Q b Q b
Qj=1Qk=1 − b
so that Q Q
(x;a+nb) (x;nb) (x;a) mod φ (x)Z[x]. (3.2)
b
Q ≡ Q Q
We shall prove that this congruence also holds when a/b belongs to . Indeed, in this
case we have ∆(a/b) 1 by assumption. By (3.1), the φ (x)-valuationDof (x;a+nb) is
b
∆(a + n) = ∆(a/b) ≥ 1, and the φ (x)-valuation of (x;a) is also ∆(a/Qb) 1. Hence
b ≥ b Q ≥
both polynomials are divisible by φ (x) and (3.2) holds.
b
Now we shall prove that
(x;nb) (1;n) mod φ (x)Z[x]. (3.3)
b
Q ≡ Q
We have
u ei·nb(1 xk) u ei·n(1 xkb) b−1 u ei·n−1(1 xℓ+kb)
(x;nb) = i=1 k=1 − = i=1 k=1 − i=1 k=1 −
Q v fj·nb(1 xk) v fj·n(1 xkb) × v fj·n−1(1 xℓ+kb) ·
Qj=1Qk=1 − Qj=1Qk=1 − Yℓ=1 Qj=1Qk=1 −
From e = fQ, we aQlso derive Q Q Q Q
| | | |
u ei·n(1 xkb) u ei·n 1−xkb
i=1 k=1 − = i=1 k=1 1−xb ,
v fj·n(1 xkb) v fj·n 1−xkb
Qj=1Qk=1 − Qj=1Qk=1 1−xb
which is a rational fractiQon wiQthout pole at x =Qξ anQd whose value at ξ equals (1;n).
b b
Q
Furthermore, for each ℓ in 1,...,b 1 , we have
{ − }
u ei·n−1(1 ξℓ+kb)
i=1 k=1 − b = (1 ξℓ)(|e|−|f|)·n = 1.
v fj·n−1(1 ξℓ+kb) − b
Qj=1Qk=1 − b
Since (x;nb) Z[x]Qand Q(ξ ;nb) = (1;n), we obtain (3.3) as expected.
b
Q ∈ Q Q (cid:3)
6
We now show how Proposition 1.4 yields on the one hand congruences through a spe-
cialization rule, and on the other hand algebraic independence results.
Recall that, under the conditions of Proposition 1.4, we have
F (q;x) = (q;n)xn Z[q][[x]].
e,f e,f
Q ∈
nX∈Nd
Then the congruence relation in Proposition 1.4 is equivalent to
F (q;x) A(q;x)F (1;xb) mod φ (q)Z[q][[x]]
e,f e,f b
≡
for every positive integer b andwith theadditional conditionthat A(q;x) inZ[q][x] satisfies
deg A(q;x) b 1 for all i, 1 i d. The following proposition is the key to prove
xi ≤ − ≤ ≤
congruences for multisums of q-factorial ratios as in Proposition 1.6.
Proposition 3.1. Assume the conditions of Proposition 1.4 are satisfied. Moreover, let
t Nd and m Nd be such that if x in [0,1)d satisfies m x 1, then ∆ (x) 1. Then,
e,f
∈ ∈ · ≥ ≥
for every positive integer b, we have:
F (q;qt1xm1,...,qtdxmd) B(q;x)F (1;xbm1,...,xbmd) mod φ (q)Z[q][[x]],
e,f e,f b
≡
where B(q;x) is a one variable polynomial in Z[q][x] satisfying deg B(q;x) b 1.
x ≤ −
Choosing e = ((2,1),(1,1)) and f = ((1,0),(1,0),(1,0),(0,1),(0,1)), we get that
[2n +n ] ![n +n ] !
F (q;x,y) = 1 2 q 1 2 q xn1yn2.
e,f [n ] !3[n ] !2
1 q 2 q
n1X,n2≥0
By Proposition 2 in [5], the function ∆ is greater than or equal to 1 on so that the
e,f e,f
D
conditions of Proposition 1.4 are satisfied. Furthermore, we can use Proposition 3.1 with
t = (t,0) and m = (1,1) which yields
F (q;qtx,x) B(q;x)F (1;xb,xb) mod φ (q)Z[q][[x],
e,f e,f b
≡
where B(q;x) is a polynomial in Z[q][x] satisfying deg B(q;x) b 1. A direct compu-
x ≤ −
tation shows that
∞ n 2
n n+k
F (q;qtx,x) = qtk xn
e,f
k k
n=0 k=0 (cid:20) (cid:21)q(cid:20) (cid:21)q
XX
and
∞ n 2
n n+k
F (1;x,x) = xn.
e,f
k k
n=0 k=0(cid:18) (cid:19) (cid:18) (cid:19)
XX
This yields the congruences for q-analogs of the first Apéry sequence a (q) given in Propo-
n
sition 1.5. The result for the second Apéry sequence b (q) is derived along the same line.
n
To prove Proposition 1.6, we first show by Proposition 3.1 that each series f(q;x) in
q
F
belongs to (q;f(1;x)). For example, we use the following specialization associated with
t = (0,0) anDd m = (1,1):
∞ n r
n
xn = F (q;x,x),
e,f
k
n=0 k=0(cid:20) (cid:21)q
XX 7
where
[n +n ] !r
F (q;x,y) = 1 2 q xn1yn2.
e,f
[n ] !r[n ] !r
1 q 2 q
n1,n2≥0
X
By Proposition 1.2 and Section 9.3 in [2], we know that (the set of all series f(1;x)) is
1
F
a subset of L and that all elements of are algebraically independent over C(x). Hence
1 1
F
Theorem 1.3 implies that, for every nonzero complex number q, all elements of are
q
F
algebraically independent over C(x).
4. Sketch of proof of Theorem 1.3
Though Theorem 1.3 holds true for all nonzero complex number q, we will focus here
on the case where q is an algebraic number. The case where q is transcendental is actually
simpler even if it requires specific considerations we do not want to deal with here for space
limitation.
Throughout this section, we fix a nonzero algebraic number q. We let K be the number
field Q(q) and R := (K) be its ring of integers. Recall that R is thus a Dedekind domain.
O
The proof of Theorem 1.3 relies on the following Kolchin-like proposition which is a
special instance of Proposition 4.3 in [2].
Proposition 4.1. Let p be a prime number, F be a finite extension of degree d of F , and
p p
k be a positive integer such that d k. Let f ,...,f be nonzero power series in F[[x]]
p 1 n
|
satisfying f (x) = A (x)f (xpk) for some A F[x] and every 1 i n. If f ,...,f
i i i i 1 n
satisfy a nontrivial polynomial relation of deg∈ree d with coefficien≤ts in≤F(x), then there
exist m ,...,m Z, not all zero, and a nonzero r(x) F(x) such that
1 n
∈ ∈
A (x)m1 A (x)mn = r(x)pk−1.
1 n
···
Furthermore, m + +m d and m d for 1 i n.
1 n i
| ··· | ≤ | | ≤ ≤ ≤
We will also need the following result which will enable us to connect reductions modulo
prime numbers and modulo cyclotomic polynomials.
Proposition 4.2. There exists an infinite set of maximal ideals of R such that, for all
S
p , we have Z[q] R and φ (q)Z[q] pR for some prime number b (depending on
p b p
∈ S ⊂ ⊂
p).
For space limitation, Proposition 4.2 will not be proved here. Its proof is elementary
when q is a root of unity and relies on the S-unit theorem otherwise. We will also need
the two following auxiliary results, the first of which being Lemma 4.4 in [2].
Lemma 4.3. Let R be a Dedekind domain, K be its field of fractions, and g ,...,g be
1 n
power series in R[[x]]. If g ,...,g are linearly dependent over R/p for infinitely many
1|p n|p
maximal ideals p, then f ,...,f are linearly dependent over K.
1 n
Lemma 4.4. Let K be a commutative field and set b a positive integer. Let r(x) K(x)
and s(x) K(x) K[[x]] be two rational fractions such that s(0) = 0. If there e∈xists a
nonzero (m∈od p if∩char(K) = p) integer m satisfying s(xb) = r(x)m,6 then there exists t(x)
in K(x) such that r(x) = t(xb).
8
Proof of Theorem 1.3. Let be the set of maximal ideals of R := (K) given in Proposi-
S O
tion 4.2. With all p in , we associate a prime number p such that the residue field R/p is
S
a finite field of characteristic p so that pZ p. Let d be the degree of the field extension
p
R/p over F . Since g belongs to , there⊂exists a polynomial A Z[x] such that
p i d i
L ∈
g (x) A (x)g (xpki) mod p[[x]]
i i i
≡
with deg A pki 1. We set k := lcm(d ,k ,...,k ). Then iterating the above relation,
xj i ≤ − p 1 n
for all i in 1,...,n and all p in , there exists B (x) in Z[x] satisfying
i
{ } S
g (x) B (x)g xpk mod p[[x]], (4.1)
i i i
≡
with degxj(Bi) ≤ pk −1. (cid:0) (cid:1)
Nowlet usassume by contradiction that f ,...,f arealgebraically dependent over C(x)
1 n
and thus over K(x) for the coefficients of the formal power series f belong to K (see for
i
instance[2]). LetQ(x,y ,...,y )beanonzeropolynomialinR[x][y ,...,y ]oftotaldegree
1 n 1 n
at most κ in y ,...,y such that Q x,f (q;x),...,f (q;x) = 0. Since f (q;g ), for
1 n 1 n i i
∈ D
every i in 1,...,n , Proposition 4.2 implies that f (q;x) A (q;x)g (xb) mod pR [[x]],
{ } (cid:0) i ≡(cid:1) i i p
for some prime b. Since Q and the series f are all nonzero and R is a Dedekind domain,
i
there thus exists an infinite subset ′ of such that, for every p in S′, the relation
S S
Q x,A (q;x)g (xb),...,A (q;x)g (xb) 0 mod pR [[x]]
1 1 n n p
≡
provides a nontri(cid:0)vial algebraic relation over Rp/p = R(cid:1)/p between the series gi|p(xb). By
(4.1), one has g (xb) B (xb)g xbpk mod p[[x]] and Proposition 4.1 then applies to
i i i
≡
g (xb),...,g (xb) by taking F = R/p. There thus exist integers m ,...,m , not all
1|p n|p (cid:0) (cid:1) 1 n
zero, and a nonzero rational fraction r(x) in F(x) such that
B (xb)m1 B (xb)mn = r(x)pk−1. (4.2)
1|p n|p
···
As g belongs to , the constant coefficient in the left-hand side of (4.2) is equal to 1. By
i d
Lemma 4.4, as pLk 1 = 0 mod p, there exists a rational fraction u(x) in F(x) such that
− 6
r(x) = u(xb) and we obtainthat B (x)m1 B (x)mn = u(x)pk−1. Furthermore, we have
1|p n|p
···
m + +m κ and m κ for 1 i n. Note that the rational fractions B , u and
1 n i i
| ··· | ≤ | | ≤ ≤ ≤
the integers m all depend on p. However, since all the integers m belong to a finite set,
i i
the pigeonhole principle implies the existence of an infinite subset ′′ of ′ and of integers
S S
t ,...,t such that, for all p in ′′, we have m = t for 1 i n. We can thus assume
1 n i i
that p belongs to ′′ and write uS(x) = s(x)/t(x) with s(x)≤and≤t(x) in F[x] and coprime.
Since degB pkS 1, the degrees of s(x) and t(x) are bounded by t + + t nκ.
i 1 n
Set h(x) := g≤(x)−−t1 g (x)−tn Z[[x]] R[[x]]. Then we obtain th|at| ··· | | ≤
1 n
··· ∈ ⊂
h xpk = g xpk −t1 g xpk −tn
|p 1|p n|p
···
= g (x)−t1 g (x)−tnB (x)t1 B (x)tn
(cid:0) (cid:1) 1|p(cid:0) (cid:1) n|p (cid:0) (cid:1)1|p n|p
··· ···
= h (x)u(x)pk−1.
|p
Since h is nonzero, we obtain that h (x)pk−1 = u(x)pk−1 and there exists a in a suitable
|p p
algebraic extension of F such that h |(x) = au(x). As the coefficients of h and u belong
|p |p
9
to R/p, we get a R/p. Thus for infinitely many maximal ideals p, the reduction modulo
p of the power se∈ries xmh(x) and xm, 1 i n, 0 m nκ, are linearly dependent
i i ≤ ≤ ≤ ≤
over R/p. Since R is a Dedekind domain, Lemma 4.3 implies that these power series are
linearly dependent over K, which means that h(x) belongs to K(x). This is a contradiction
as g ,...,g are algebraically independent over C(x). (cid:3)
1 n
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B. Adamczewski, É. Delaygue and F. Jouhet, Univ Lyon, Université Claude Bernard Lyon
1, CNRS UMR 5208, Institut Camille Jordan, F-69622 Villeurbanne Cedex, France. Emails:
[email protected], [email protected] and [email protected]
J. P. Bell, Department of Pure Mathematics, University of Waterloo, Waterloo, ON, Canada.
Email: [email protected]
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