Download THE NUMERICAL FACTORS OF ∆n(f,g)

Indian J. Pure Appl. Math., 46(5): 701-714, October 2015
c Indian National Science Academy
°
DOI: 10.1007/s13226-015-0140-9
THE NUMERICAL FACTORS OF ∆n (f, g)1
Qingzhong Ji and Hourong Qin
Department of Mathematics, Nanjing University, Nanjing 210093, P. R. China,
e-mails: [email protected], [email protected]
(Received 19 June 2014; after final revision 30 September 2014;
accepted 24 December 2014)
Let α1 , α2 , . . . , αr be the roots of the polynomial f (x) = xr + a1 xr−1 + · · · + ar ∈ Z[x]
and let g = {gn (X)}n∈N , where gn (X) = gn (x1 , x2 , . . . , xr ) ∈ Z[x1 , x2 , . . . , xr ] is a symmetric polynomial. For each n, put ∆n (f, g) = gn (α1 , α2 , . . . , αr ). In this paper, for a special
symmetric polynomial sequence g, we investigate the numerical factors of ∆n (f, g). If p is a
prime, we establish an analogue of Iwasawa’s theorem in algebraic number theory for the orders
ordp (∆npt (f, g)) of the p-primary part of ∆npt (f, g) when t varies.
Key words : Recurring series; Iwasawa theory; cyclotomic polynomial.
1. I NTRODUCTION
Throughout this paper, let Q, Z and N denote the field of rational numbers, the ring of rational integers
and the set of nonnegative integers, respectively. Let N∗ = N \ {0}. As usual, let ordp denote the
p-adic valuation of Qp such that ordp (p) = 1.
Let α1 , α2 , . . . , αr be the roots of the polynomial
f (x) = xr + a1 xr−1 + · · · + ar−1 x + ar
(1)
whose coefficients are rational integers. Suppose g = {gn (X)}n∈N is a polynomial sequence, where
gn (X) = gn (x1 , x2 , . . . , xr ) ∈ Z[x1 , x2 , . . . , xr ] is a symmetric polynomial in r variables. For each
n, put
∆n (f, g) = gn (α1 , α2 , . . . , αr ).
1
(2)
Supported by NSFC (Nos. 11171141, 11471154, 11271177), NSFJ (Nos. BK2010007, BK2010362), PAPD and the
Cultivation Fund of the Key Scientific and Technical Innovation Project, Ministry of Education of China (No. 708044).
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QINGZHONG JI AND HOURONG QIN
It is clear that ∆n (f, g) ∈ Z. For example, if gn (x1 , x2 , . . . , xr ) =
∆n (f, g) =
Qr
i=1 (1
− xni ), then
r
Y
(1 − αin ) ∈ Z.
(3)
i=1
This function was introduced by Pierce [4] who studied the forms of its primitive factors. Later,
Lehmer [3] made a detailed study of this sequence of numbers.
The theory of Zp -extensions is one of the most fruitful areas of research in number theory. A
beautiful result in this area is the theorem of Iwasawa which describes the behavior of the p-part of
the class number in a Zp -extension of number fields.
Iwasawa Theorem ([7], Theorem 13.13.) — Let K∞ /K be a Zp -extension and K∞ =
with [Kn : K] =
pn .
Let
pen
S+∞
n=0 Kn
be the exact power of p dividing the class number of Kn . Then there
exist integers λ ≥ 0, µ ≥ 0 and ν, all independent of n, and an integer n0 such that
en = λn + µpn + ν, f or all n ≥ n0 .
By the structure of Λ-modules, one sees that pen is indeed the value of the characteristic polynomial of some Λ-module at special points. From this viewpoint, in [2], the authors prove an analogue
of Iwasawa’s theorem for higher K-groups of curves over finite fields. Let X be a smooth projective
curve of genus g over a finite field F with q elements. For m ≥ 1, let Xm be the curve X over the
finite field Fm , the m-th extension of F. For 1 ≤ i ≤ 2g, denote by πi the characteristic roots of the
Frobenius endomorphism φ. Set
2g
Y
gn,m (x1 , x2 , · · · , x2g ) =
(1 − (q n xi )m ).
i=1
We have ]K2n (Xm ) = gn,m (π1 , π2 , · · · , π2g ), where K2n (Xm ) is the K-group of the smooth
projective curve Xm . Let p be a prime. Denote the p-primary part of the order of K2n (Xpt ) by
pen,p (t) , i.e., en,p (t) = ordp (]K2n (Xpt )).
Theorem 1.1 ([2]) — There exist integers λn,p ≥ 0, νn,p and a positive integer Tn,p such that
en,p (t) = λn,p t + νn,p , f or all t ≥ Tn,p .
Let f (x) = x2 − P x − Q ∈ Z[x] and g = {gn (x1 , x2 )}n∈N , where g0 (x1 , x2 ) = 0, gn (x1 , x2 )
Pn−1 k n−1−k
=
, n ≥ 1. It is well-known that the recurring series ∆n (f, g) is the Lucas sek=0 x1 x2
quences Ln with parameters P and Q. The following results are consequences of the well-known
properties of the Primitive Divisor Theorem for Lucas sequences.
NUMERICAL FACTORS OF ∆n (f, g)
703
Theorem 1.2 — ([1]) Let n ∈ N∗ and p a prime.
(1) If p - Ln Lp , then ordp (Lnpt ) = 0, for all t ∈ N.
(2) If p|Ln Lp , then there exist integers νn,p and Tn,p such that
ordp (Lnpt ) = t + νn,p , f or all t ≥ Tn,p .
(3) Let n ∈ N∗ and p, q be two different primes. Then there exists a positive integer Tn,p,q such
that
ordq (Lnpt ) = ordq (LnpTn,p,q ), for all t ≥ Tn,p,q ,
i.e., the numbers ordq (Lnpt ) are stable when t is sufficiently large.
(4) Let Sn,p (t) be the set of all primes which divide Lnpt . Then ]Sn,p (t) −→ +∞ as t −→ +∞.
In this paper, we generalize Pierce’s recurring series {∆n }n∈N defined by (3) and
{gn,m (π1 , π2 , · · · , π2g )}m∈N defined above to the recurring series {∆n,m }n∈N for any integer
Q
m ∈ N∗ , where ∆n,m = ri=1 (mn − αin ). We give a detailed study of essential and characteristic factors of ∆n,m especially as regards sequences of numbers. Let p be a prime and n, m ∈ N∗ , we
establish an analogue of Iwasawa’s theorem for the orders ordp (∆npt ,m ) as follows.
Theorem 3.7 — Let p be a prime. Fix integers n, m ∈ N∗ , let pen,m,p (t) be the p-primary part of
∆npt ,m for t ∈ N.
(1) If p - ∆n,m , then en,m,p (t) = 0 for all t ∈ N.
(2) If p|∆n,m , then there exist integers λn,m,p ≥ 1 and νn,m,p , both independent of t, and an
integer Tn,m,p such that
en,m,p (t) = λn,m,p t + νn,m,p , f or all t ≥ Tn,m,p .
On the other hand, let p, q be two different primes, we prove that the numbers ordq (∆npt ,m ) are
stable when t is sufficiently large (See Theorem 3.9). We also prove that the number of prime factors
of ∆npt ,m goes to infinity as t goes to infinity. (See Corollary 3.10).
2. FACTORIZATION OF ∆n,m
(m)
Let the notation be as in §1. In this section, fix an integer m ∈ N∗ , define gm = {gn (X)}n∈N as
follows
gn(m) (x1 , x2 , . . . , xr ) =
r
Y
(mn − xni ).
i=1
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QINGZHONG JI AND HOURONG QIN
2.1 Definition of ∆n,m — Let α1 , . . . , αr be the roots of the polynomial f (x) ∈ Z[x] defined by
(1). Then ∆n,m is defined by
∆n,m = ∆n (f, gm ) = gn(m) (α1 , α2 , . . . , αr ) =
r
Y
(mn − αin ).
(4)
i=1
Pierce [4] and Lehmer [3] listed many properties of ∆n = ∆n,1 . In this section, we will generalize
all results in [3] concerning ∆n to the case ∆n,m , for all n, m ∈ N∗ . We would like point out that the
idea used here is similar to that in [3].
Remark 2.2 : (1) The polynomial f (x) can be viewed as a characteristic polynomial of some r × r
matrix A, for example,





A=





0 0 ··· 0
−ar

1 0 · · · 0 −ar−1 


0 1 · · · 0 −ar−2 
.

.. ..
..
..

. .
.
.

0 0 · · · 1 −a1
Then f (x) = |xE − A| and
∆n,m = |mn E − An |,
(5)
where |B| = det(B) for any square matrix B.
(2) Let α be a root of f (x). If f (x) is irreducible, then
∆n,m = NK/Q (mn − αn ),
where K = Q(α) and NK/Q is the norm map from the field K to Q.
(3) Since (m, ar )|∆n,m , for our purposes, we always assume (m, ar ) = 1 in this section.
2.3 Essential and characteristic factors of ∆n,m
Let Φδ (x, y) be the δth homogeneous cyclotomic polynomial, i.e.,
Φδ (x, y) =
δ
Y
(x − yζδi )
(6)
i=1
(i,δ)=1
where ζδ is a primitive δth root of unity. Then we define the integer Φ∗δ,m by
Φ∗δ,m =
r
Y
i=1
Φδ (m, αi ).
(7)
NUMERICAL FACTORS OF ∆n (f, g)
It follows from the formula xn − y n =
Q
δ|n Φδ (x, y)
∆n,m =
Y
705
that
Φ∗δ,m .
(8)
δ|n
This gives a partial factorization of ∆n,m into integer factors. If we assume that each ∆, whose
first subscript is a proper divisor of n, has been factored, the complete factorization of ∆n,m depends
only on that of Φ∗n,m . For this reason we call this latter number the essential factor of ∆n,m . On the
other hand, we may consider the prime factors of ∆n,m . Similarly, the prime factors of ∆n,m which
do not divide ∆d,m , where d is a proper divisor of n, are called the characteristic prime factors of
∆n,m . The concepts of essential factor and characteristic prime factors were introduced by Lehmer
[3].
Lemma 2.4 — The essential factor Φ∗n,m of ∆n,m contains all the characteristic prime factors of
∆n,m .
P ROOF : By (8) a characteristic prime factor p of ∆n,m must divide Φ∗δ,m for some divisor δ of n.
If δ were less than n, and hence p would divide ∆δ,m , contrary to the definition of p. Therefore δ = n
2
and the lemma follows.
Lemma 2.5 — A characteristic prime factor p of ∆n,m cannot divide n.
P ROOF : If possible, let n = pδ. Suppose f (x) = |xE − A| for some matrix A. Then by the
multinomial theorem modulo p and (5), we have
0 ≡ ∆n,m ≡ ∆pδ,m
≡ |mpδ E − Apδ |
≡ |mδ E − Aδ |p
≡ |mδ E − Aδ |
≡ ∆δ,m
(mod p).
This contradicts the hypothesis that p is a characteristic factor of ∆n,m .
2
Remark : (1) It is not true that the essential factor of ∆n,m is made up exclusively of characteristic
prime factors (See [3], p. 462).
(2) The essential factor Φ∗n,m may, however, have a factor in common with n.
(3) If f is reducible over the rational field so that f = f1 f2 , then
∆n,m (f ) = ∆n,m (f1 )∆n,m (f2 )
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QINGZHONG JI AND HOURONG QIN
is a factorization into integers. Hence for our purposes we may suppose that f is irreducible. The
following result is a generalization of [7], Lemma 2.9.
Theorem 2.6 — If pe (e > 0) is the highest power of a characteristic prime factor p of ∆n,m (f ),
where f is irreducible and of degree r, and if w is the order of p mod n, then w ≤ r and e is divisible
by w.
2
P ROOF : It is similar to the proof of [3], Theorem 3.
2.7 The recurring series for ∆n,m
In order to render the factorization of ∆n,m practical, it is first necessary to have a simple method of
calculating its actual value. This is done with the help of a polynomial Mm (x) uniquely determined
by f (x) and m in the following manner. Let
f0 (x) = x − mr ,
Qr
(x − mr−1 αi ),
f1 (x) = i=1
Q
f2 (x) = 1≤j<i≤r (x − mr−2 αi αj ),
···
···························
fr (x) = x − α1 α2 · · · αr = x − (−1)r ar .
Mm (x) is defined as the least common multiple of these f 0 s. That is,
Mm (x) = [f0 (x), f1 (x), . . . , fr (x)] = xq + A1 xq−1 + A2 xq−2 + · · · + Aq .
Definition 2.8 — Let {xn }∞
n=0 be a recurring series such that
xn+e + a1 xn+e−1 + · · · + ae−1 xn+1 + ae xn = 0, for all n ≥ 0.
Then the polynomial xe + a1 xe−1 + · · · + ae−1 x + ae is called the scale of the recurring series
{xn }∞
n=0 .
Theorem 2.9 — The numbers
∆0,m , ∆1,m , ∆2,m , ∆3,m , · · ·
form a recurring series whose scale is Mm (x). That is, for every n ≥ 0,
∆n+q,m + A1 ∆n+q−1,m + · · · + Aq−1 ∆n+1,m + Aq ∆n,m = 0.
P ROOF : By the definition (4) of ∆n,m , we have
∆n,m
r
r
X
Y
n
n
(m − αi ) =
=
i=1
X
k=0 1≤i1 <···<ik ≤r
(−1)k m(r−k)n αin1 · · · αink .
(9)
NUMERICAL FACTORS OF ∆n (f, g)
707
Hence we obtain
q
X
Ai ∆n+q−i,m =
i=0
r
X
X
(−1)k m(r−k)n αin1 · · · αink Mm (m(r−k) αi1 · · · αik ) = 0,
k=0 1≤i1 <···<ik ≤r
where A0 = 1. This completes the theorem.
2
2.10 q-periodic ∆’s
Lehmer [3] has proved that ∆n,1 is a periodic function of proper period τ if and only if f (x)
= Φτ (x, 1). In this subsection, for a fixed integer m, we will consider the periodic properties of
∆n,m .
Definition 2.11 — Suppose F : Z −→ C is a number theory function. We call F a q-periodic
function of period τ, if there exists a function λ(n) such that
F (qτ + k) = λ(q)F (k), for all q, k ∈ Z.
(10)
The function λ is called a periodic factor of F. We also call F q-periodic with respect to λ. A
positive integer τ is called a proper period of a q-periodic function F, if for any positive integer T < τ,
F is not q-periodic of period T.
Remark 2.12 : (1) It is obvious that a periodic function F is q-periodic, in this case
λ(n) = 1, for all n ∈ Z.
(2) If the function F is defined over N and τ is a positive integer such that
F (qτ + k) = λ(q)F (k), for all q, k ∈ N,
for some function λ(n) defined over N, then F can be extended to a q-periodic function defined over
Z. In this case we also call F a q-periodic function defined over N.
Lemma 2.13 — If F 6= 0 is a q-periodic function with respect to a function λ, then λ(n) 6= 0 for
all n ∈ Z and λ : Z −→ C∗ is a group homomorphism.
P ROOF : Suppose there exists an integer n0 ∈ Z such that λ(n0 ) = 0. By (10), for any n ∈ Z, we
have
F (n) = F (n0 T + (n − n0 T )) = λ(n0 )F (n − n0 T ) = 0.
This contradicts the assumption F 6= 0. It is easy to see that λ(0) = 1 and λ(m+n) = λ(m)λ(n)
for all m, n ∈ Z. Hence λ is a group homomorphism.
2
Lemma 2.14 — Suppose τ is a proper period of a q-periodic function F. If T is a period of F,
then τ |T.
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QINGZHONG JI AND HOURONG QIN
P ROOF : First we prove that q-periodic functions have properties similar to those of periodic
functions. Let T > 0 be a period of a q-periodic function F (n), i.e., there exists a function λ(n) such
that
F (qT + k) = λ(q)F (k), for all q, k ∈ Z.
Then we have
(i) for any a ∈ Z, aT is a period of F. In fact, set λa (n) = λ(an), then
F (qaT + k) = λ(aq)F (k) = λa (q)F (k), for all q, k ∈ Z.
(ii) if T1 and T2 are two periods of F, then T1 + T2 is also a period of F. Assume λi is corresponding to Ti , i = 1, 2. Set λ(n) = λ1 (n)λ2 (n), then
F (q(T1 + T2 ) + k) = λ1 (q)F (qT2 + k) = λ1 (q)λ2 (q)F (k) = λ(q)F (k), for all q, k ∈ Z.
Suppose τ - T. Then T = q0 τ + b where q0 , b ∈ Z and 0 < b < τ. By (i) and (ii) above, we
obtain that b = T − q0 τ is a period of F. This contradicts the fact that τ is a proper period of F.
2
For a fixed integer m, it may happen that ∆n,m is a λ-periodic function of n. In this case we have
Theorem 2.15 — A necessary and sufficient condition for ∆n,m to be q-periodic function of n
of proper period τ is that f (x) = Φτ (x, m), where Φτ (x, y) is the τ -th homogeneous cyclotomic
polynomial defined by (6).
P ROOF : If ∆n,m is q-periodic of proper period τ , then ∆τ,m = 0. Hence f has a root α for which
ατ = mτ . Then there exists a primitive kth root ζk of unity such that α = mζk . Since f is irreducible
all its roots are mζki , 1 ≤ i ≤ k, (k, i) = 1, so that f (x) = Φk (x, m), where k is some divisor of τ.
But ∆n,m is of period k, for if n, j are any integers ≥ 0,
∆nk+j,m =
Y
Y
(mnk+j − αink+j ) = mϕ(k)kn (mj − αij ) = λ(n)∆j,m ,
i
i
where λ(n) = mϕ(k)kn is the periodic factor. Hence by Lemma 2.14, τ is a divisor of k. Therefore
τ = k and f (x) = Φτ (x, m).
2
3. I WASAWA T HEORY OF ∆n,m
Let the notation be as in §2. For our purposes, in this section, we make the following hypothesis:
(H 1) f (x) is defined by (1) and irreducible.
(H 2) Fix an integer m satisfying (m, ar ) = 1.
NUMERICAL FACTORS OF ∆n (f, g)
709
(H 3) f (x) 6= ΦT (x, m) for all T ∈ N∗ .
Let K = Q(α1 , α2 , . . . , αr ) be the splitting field of f (x) over the rational number field Q, OK
the ring of algebraic integers of K. For any prime p, let P be a prime ideal of K lying above p.
Theorem 3.1 — Let n ∈ N and p be a prime factor of ∆n,m . Then, for any positive integer t
∆
.
satisfying p|t, we have p| ∆nt,m
n,m
P ROOF : By the formula (4), the condition p|∆n,m implies mn ≡ αin (mod P) for some i(1 ≤
i ≤ r). If p|t, then we have
r
r X
t−1
Y
Y
mnt − αjnt
∆nt,m
n(t−1−k)
n(t−1)
=
≡
tm
mnk αj
≡ 0 (mod P).
∆n,m
mn − αjn
j=1
j=1
j6=i
k=0
∆
Hence p| ∆nt,m
.
n,m
2
Corollary 3.2 — (1) Let n ∈ N∗ and p a prime factor of ∆n,m . Then, for all t ∈ N, we have
pe+t |∆npt ,m , where e = ordp (∆n,m ).
(2) Let n, t ∈ N∗ . Then we have (∆n,m )t |∆n(∆n,m )t−1 ,m .
P ROOF : (1) It follows easily by induction on t. (2) It follows trivially from (1) and the fact:
∆n1 ,m |∆n2 ,m , if n1 |n2 .
2
Theorem 3.1 is about divisibility. The next result will be about non-divisibility. First, a definition.
If p is a prime, put d(p) = lcm1≤i≤[K:Q] {pi − 1}.
2
Theorem 3.3 — Let n, t ∈ N∗ . Suppose p is a prime such that p - ∆n,m and (t, d(p)) = 1. Then
(i) p - ∆nt,m ; (ii) p - ∆npx ,m for any x ∈ N.
P ROOF : It is clear that (ii) follows (i). Hence it suffices to prove (i). If p|∆nt,m , then
mnt ≡ αint
(mod P) for some i (1 ≤ i ≤ r).
(11)
If αi ≡ 0 (mod P), then m ≡ 0 (mod P). Hence p|(ar , m), this contradicts the assumption
(ar , m) = 1. So m, αi 6∈ P. By (11), we have
µ n ¶t
αi
≡1
mn
αn
(mod P).
αn
But ( min )d(p) ≡ 1 (mod P) and (t, d(p)) = 1, we have min ≡ 1 (mod P), i.e., mn ≡ αin
Q
(mod P). Hence ∆n,m = rj=1 (mn − αjn ) ≡ 0 (mod P) contradicts p - ∆n,m . Hence p - ∆nt,m .
This completes the proof.
2
710
QINGZHONG JI AND HOURONG QIN
Lemma 3.4 — Let n be the smallest integer such that p|∆n,m . Then n|d(p).
P ROOF : From the proof of Theorem 3.3, if p|∆n,m , then there exists an index i (1 ≤ i ≤ r) such
that mn ≡ αin (mod P) and m, αi 6∈ P.
(i) Assume m ≡ αi (mod P). Then ∆1,m =
(ii) Assume m 6≡ αi (mod P). Then
(
αi
m
Qr
j=1 (m
− αj ) ≡ 0 (mod P), so n = 1.
6≡ 1 (mod P) and
αi n
) ≡ 1 (mod P).
m
From the definition of n, it follows that n is the order of
( αmi )d(p)
αi
m
(mod P). On the other hand,
≡ 1 (mod P). Hence n|d(p) as asserted.
2
Corollary 3.5 — Let p be a prime. Then p|∆1,m if and only if m ≡ αi (mod P) for some
i (1 ≤ i ≤ r).
Let Qp be the p-adic completion of Q. Let Q and Qp be the algebraic closures of Q and Qp ,
respectively. Let ρ be an embedding of Q into Qp . We simply rename ρ(a) as a.
S
We will keep the notation ordp for the additive valuation from Qp to Q {∞}, extended by the
S
standard additive valuation ordp from Qp to Z {∞}, namely, if α ∈ Qp , then
ordp (α) = [Qp (α) : Qp ]−1 ordp (NQp (α)/Qp (α)).
Here NQp (α)/Qp is the usual norm map from Qp (α) to Qp .
Lemma 3.6 ([6], p. 172-174) — Let p and q be different primes. For n ≥ 1, let ξ ∈ Qp be any
primitive pn -th root of unity. Then the following results hold.
(1) ordp (ξ − 1) =
1
pn−1 (p−1)
and ordq (ξ − 1) = 0.
(2) Let α ∈ Qp be integral over Zp .
t
(i) If ordp (α − 1) = 0, then ordp (αp − 1) = 0 for all positive integers t ≥ 1.
(ii) If ordp (α − 1) > 0, then there exist an integer t0 and a constant c depending on α such that
t
ordp (αp − 1) = t + c,
for all t ≥ t0 . In fact, t0 and c can be chosen as
t0 = min{t ∈ Z|
1
pt−1 (p
− 1)
< ordp (α − 1)},
NUMERICAL FACTORS OF ∆n (f, g)
and
c = ordp (α − 1) +
X
711
[ordp (α − ξ) − ordp (1 − ξ)],
16=ξ∈S
where S is the set of
pi -th
roots of unity, 1 ≤ i < t0 .
(3) Let β ∈ Qq be integral over Zq .
t
(i) If ordq (β − 1) > 0, then ordq (β p − 1) = ordq (β − 1) > 0, for all t ≥ 1.
(ii) If ordq (β − 1) = 0, then there exists an integer t0 ≥ 0 such that, for all t ≥ t0 ,
t
t0
ordq (β p − 1) = ordq (β p − 1).
Let n ∈ N∗ and p a prime. Set en,m,p (t) = ordp (∆npt ,m ) for t ∈ N.
Theorem 3.7 — Let n ∈ N∗ and p a prime.
(1) If p - ∆n,m , then en,m,p (t) = 0 for all t ∈ N.
(2) If p|∆n,m , then there exist integers λn,m,p ≥ 1, νn,m,p and Tn,m,p such that
en,m,p (t) = λn,m,p t + νn,m,p , f or all t ≥ Tn,m,p .
P ROOF : By Theorem 3.3, if p - ∆n,m , then en,m,p (t) = 0, for all t ∈ N. Hence, it suffices to
prove (2). Assume p|∆n,m . Without loss of generality, we may assume n is the smallest integer such
that p|∆n,m . By Lemma 3.4, n|d(p), hence (p, n) = 1. Let ζn be a primitive nth root of unity. Then
∆n,m =
r
r n−1
r n−1
Y
Y
Y
Y
Y
αi
(mn − αin ) =
(m − αi ζnj ) = mrn
(1 − ζnj ).
m
i=1
i=1 j=0
i=1 j=0
Note that p|∆n,m implies ordp (m) = 0 (see the proof of Theorem 3.3). Hence we have
ordp (1 −
αi j
m ζn )
≥ 0 for all 1 ≤ i ≤ r, 0 ≤ j ≤ n − 1. For each i (1 ≤ i ≤ r), we claim
αi j0
m ζn ) > 0. In fact, if there
0 and ordp (1− αmi ζnj2 ) > 0, then ordp (αi ) = 0
that there is at most one index j0 (0 ≤ j0 ≤ n − 1) such that ordp (1 −
exist 0 ≤ j1 < j2 ≤ n−1 such that ordp (1− αmi ζnj1 ) >
and
ordp (1 − ζnj2 −j1 ) = ordp (
αi
αi
αi j 1
ζn (1 − ζnj2 −j1 )) = ordp ((1 − ζnj2 ) − (1 − ζnj1 )) > 0.
m
m
m
This contradicts (p, n) = 1.
Set
λn,m,p = ]{i | 1 ≤ i ≤ r, there exists an index j such that ordp (1 −
αi j
ζ ) > 0}.
m n
712
QINGZHONG JI AND HOURONG QIN
The condition p|∆n,m implies λn,m,p ≥ 1. On the other hand, we have
∆npt ,m =
=
=
=
Qr
t
npt − αnp )
i=1 (m
i
Qr Qn−1 pt
pt j
i=1
j=0 (m − αi ζn )
Q
t Q
αi p t j
mrnp ri=1 n−1
j=0 (1 − ( m ) ζn )
Qn−1
t Q
t
mrnp ri=1 j=0
(1 − ( αmi ζnj )p )
(12)
since (n, p) = 1. By (2) of Lemma 3.6, there exist integers νn,m,p and Tn,m,p such that for all
t ≥ Tn,m,p , we have
en,m,p (t) = ordp (∆npt ,m )
P Pn−1
t
= ri=1 j=0
ordp (1 − ( αmi ζnj )p )
= λn,m,p t + νn,m,p ,
where the integers λn,m,p and νn,m,p are independent of t.
2
Q
Remark 3.8 : For each n ∈ N, set fn (x) = ri=1 (x − αin ). Let p be a prime factor of ∆n,m .
Factor fn (x) over Fp [x] as follows:
fn (x) = p1 (x)e1 p2 (x)e2 · · · ps (x)es
where p1 (x), p2 (x), . . . , ps (x) ∈ Fp [x] are non-associate irreducible polynomials with multiplicity
ei ≥ 1 (1 ≤ i ≤ s). If e1 = e2 = · · · = es = 1, then λn,m,p = 1, i.e., there exists a unique index i0
such that ordp (mn − αin0 ) > 0. In fact, if there exist 1 ≤ i < j ≤ r such that ordp (mn − αin ) > 0
and ordp (mn − αjn ) > 0, then ordp (m) = 0 and so
ordp (αin − αjn ) = ordp ((mn − αjn ) − (mn − αin )) > 0.
Hence αin is a root of fn (x) with multiplicity at least 2 over Fp which is the algebraic closure of
Fp . This contradicts the assumptions e1 = e2 = · · · = es = 1.
Theorem 3.9 — Let n ∈ N∗ and p, q be two different primes. Then there exists a positive integer
Tn,m,p,q such that
ordq (∆npt ,m ) = ordq (∆npTn,m,p,q ,m ), f or all t ≥ Tn,m,p,q ,
i.e., the numbers ordq (∆npt ,m ) are stable when t is sufficiently large.
P ROOF : Without loss of generality, we may assume (n, p) = 1 and ordq (m) = 0. On the other
hand, by (12), we have
rnpt
∆npt ,m = m
r n−1
Y
Y
αi
t
(1 − ( ζnj )p ).
m
i=1 j=0
NUMERICAL FACTORS OF ∆n (f, g)
713
(i)
(i)
For each i (1 ≤ i ≤ r), we divide the set {j|0 ≤ j ≤ n − 1} = S1 ∪ S2 in such a way
(i)
(i)
t
pj
(i)
that for each j ∈ S1 , there is a tj ≥ 0 such that ordq (1 − ( αmi ζnj )
(i)
) > 0, and for j ∈ S2 , the
(i)
equality ordq (1 − ( αmi ζnj )p ) = 0 holds, for all t ≥ 0. Set Tn,m,p,q = max 1≤i≤r {tj }. Then, for all
t
(i)
j∈S1
t ≥ Tn,m,p,q , by Lemma 3.6, we have
ordq (∆npt ,m ) =
=
Pr
Pn−1
Pr
P
i=1
i=1
j=0
ordq (1 − ( αmi ζnj )p )
t
(i)
j∈S1
ordq (1 −
(i)
t
j
( αmi ζnj )p
).
Since the last sum does not depend on t, the result follows.
2
Corollary 3.10 — Let Sn,m,p (t) be the set of all primes which divide ∆npt ,m . Then
]Sn,m,p (t) −→ +∞ as t −→ +∞.
P ROOF : Suppose that there exists integer t0 such that for all t ≥ t0 , Sn,m,p (t) = Sn,m,p (t0 ).
By Theorem 3.7 and Theorem 3.9, it would follow that ∆npt ,m would be equal to a constant times
pλn,m,p t for large t, i.e., there exist positive constant numbers T and c such that
|∆npt ,m | = cpλn,m,p t
(13)
for all t ≥ T. On the other hand, the assumption (H3) implies that m 6= |αi |, 1 ≤ i ≤ r. Set
S1 = {i | m > |αi |}, S2 = {i | m < |αi |}, b =
Y
max{|m|n , |αi |n }.
1≤i≤r
If m = 1, then S2 6= ∅. Hence, for all m ≥ 1, we have b > 1 and
lim
t−→+∞
|∆npt ,m |
bpt
Q
=
=
=
t−→+∞
t
1≤i≤r
lim
bpt
t
t
|mnp −αnp
| Q
|mnp −αnp
|
i
i
· i∈S2
t
i∈S
np
1
m
|αi |npt
t−→+∞
Q
Q
t
t
lim
|1 − ( αmi )np | · i∈S2 |1 − ( αmi )np |
i∈S
1
t−→+∞
lim
Q
t
|mnp −αnp
|
i
t
t
= 1.
t
Therefore for sufficiently large t, we have |∆npt ,m | > abp for some constant a > 0. Clearly, this
is incompatible with (13) just given.
2
At last, we give the following definition.
Definition 3.11 — A sequence of integers {an } is called an Iwasawa sequence if for any positive
integer m and prime p, there exist integers λ, T ∈ N and ν ∈ Z such that
ordp (ampt ) = λt + ν, f or all t ≥ T.
714
QINGZHONG JI AND HOURONG QIN
Example 3.12 : Let m be any positive integer. Then the sequence of binomial coefficients
{Cnm }n≥m
is an Iwasawa sequence. In fact, by Kummer Theorem,
m
ordp (Cnp
t ) = t + ν, f or all t ≥ T,
where T = max{0, [logp m] − ordp (n) + 1} and ν = ordp (n) − ordp (m).
ACKNOWLEDGEMENT
We would like to thank the referee for reading the manuscript carefully and providing valuable comments and suggestions.
R EFERENCES
1. Yu. F. Bilu, G. Hanrot and P. M. Voutier, Existence of primitive divisors of Lucas and Lehmer numbers,
J. reine angew Math., 539 (2001), 75-122.
2. Q. Ji and H. Qin, Higher K-groups of smooth projective curves over finite fields, Finite Fields and Their
Applications, 18 (2012), 645-660.
3. D. H. Lehmer, Factorization of certain cyclotomic funtions, Annals of Math., 34(2) (1933), 461-479.
Qn
4. T. A. Pierce, The numerical factors of the arithmetic forms i=1 (1 ± αim ), Annals of Math., 18 (1916),
53-64.
5. P. Ribenboim, My Numbers, My Friends, Springer-Verlag 2000.
6. M. Rosen, Number Theory in Function Fields, GTM 210, Springer-Verlag, New York, 2002.
7. L. C. Washington, Introduction to cyclotomic fields (Second Edition), GTM 83, Springer-Verlag, New
York, 1996.