Continued fraction algorithms and Lagrange's theorem in ${\mathbb Q}_p$

0.2 MB·

by  Asaki Saito

Continued fraction algorithms and Lagrange’s
7
theorem in Q
1 p
0
2
n Asaki Saito1,∗, Jun-ichi Tamura2, and Shin-ichi Yasutomi3
a
J 1Future University Hakodate, Hakodate, Hokkaido 041-8655, Japan

7
2Tsuda College, Kodaira, Tokyo 187-8577, Japan
1
3Toho University, Funabashi, Chiba 274-8510, Japan

]
T
N January 18, 2017
.
h
t
a
m
Abstract
[
We present several continued fraction algorithms, each of which gives

1 aneventuallyperiodicexpansionforeveryquadraticelementofQ overQ
p
v and gives a finite expansion for every rational number. We also give, for
5 each of our algorithms, the complete characterization of elements having
1
purely periodic expansions.
6
4
0 1 Introduction
.
1
0 Inthispaper,weintendtoaddaclassicalflavortothep-adicworldrelatedtothe
7
well-knowntheoremofLagrange(resp.,Galois)onthecompletecharacterization
1
of eventually (resp., purely) periodic continued fractions (cf. [5], [4]; see also
:
v [7]).
i
X In what follows, p denotes a prime, Qp the field of p-adic numbers, and
Z the ring of p-adic integers. Schneider [9] gave an algorithm that generates
r p
a continued fractions of the form
pk1
(k ∈Z ,k ∈Z ,d ∈{1,...,p−1} (n≥1))
pk2 1 ≥0 n+1 >0 n
d +
1 pk3
d +
2
...
and found periodic continued fractions for some quadratic elements of Z over
p
Q(seealso[2]). Ruban[8]gaveanalgorithmthatgeneratescontinuedfractions
∗Authortowhomcorrespondenceshouldbeaddressed. Electronicmail: [email protected]
1
of the shape
1
g + ,
0
1
g +
1
1
g +
2
...
where
0
g ∈ e pi m∈Z ,e ∈{0,1,...,p−1} (n≥0).
n i ≥0 i
( (cid:12) )
i=X−m (cid:12)
(cid:12)
(cid:12)
On the other hand, Weger(cid:12)[15] has found a class of infinitely many quadratic
elements α ∈ Q over Q such that the continued fraction expansion of α ob-
p
tained by Schneider’s algorithm is not periodic. Ooto [6] has found a similar
result related to the algorithm given by Ruban. Weger [16] has mentioned in
his paper, “it seems that a simple and satisfactory p-adic continued fraction
algorithm does not exist”, and given a periodicity result of lattices concern-
ing quadratic elements of Q . Browkin [1] has proposed some p-adic continued
p
fraction algorithms; nevertheless, the periodicity has not been proved for the
continued fractions obtained by applying his algorithms to quadratic elements
of Q . By disclosing a link between the hermitian canonical forms of certain
p
integral matrices and p-adic numbers, Tamura [10] has shown that a multi-
dimensional periodic continued fraction converges to α,α2,...,αn−1 in the
p-adic sense without considering algorithms of continued fraction expansion,
(cid:0) (cid:1)
where α is the root of a polynomial in Z[X] of degree n stated in Lemma 4.1.
We can summarize the above situation as follows: There have been proposed
several p-adic continued fraction algorithms. However,it remains quite unclear
whetherornotthereexistsasimplealgorithmthatgeneratesperiodiccontinued
fractions for all the algebraic elements of Q of fixed degree greater than one.
p
Even for quadratic elements, p-adic versions of Lagrange’s theorem have not
been found.
The main objectives of this paper are to define some algorithms generating
continued fractions of the form
t1pk1
d + (k ∈Z , t ,d ∈Z\pZ (n≥1)) (1)
0 t2pk2 n >0 n n
d +
1 t3pk3
d +
2
...
with d ∈Q such that d = [d ] (see (2) for the definition of the integral part
0 p 0 0
[α] of α∈Q ) and to give
p
(i) p-adic versions of Lagrange’s theorem for the three algorithms, i.e., the
periodicity of the resulting continued fractions for quadratic elements of
Q over Q, and
p
2
(ii) p-adic versions of Galois’ theorem concerning purely periodic continued
fractions.
Moreover, we show that the continued fraction expansions of an arbitrary ra-
tional number always terminate by our algorithms.
Itisworthmentioningthatouralgorithmshaveacommonbackgroundwith
those proposed in [11, 12, 13, 14, 3] in the design of continued fraction algo-
rithms.
The rest of this paper is organized as follows. In Section 2, we consider
expanding α ∈ Q into continued fractions whose form is more general than
p
the form (1). In Section 3, we establish convergence properties of the contin-
ued fractions introduced in Section 2. In Section 4, we give two basic maps T
i
(i = 1,2) and present related lemmas. We define three algorithms in terms of
these basic maps in Section 5. In Section 6, we show that each of our algo-
rithms gives an eventually periodic expansion for every quadratic Hensel root,
i.e., every quadratic element of Q over Q whose existence is guaranteed by
p
Hensel’s Lemma. We do the same for every quadratic element in Section 7. In
Sections 8 and 9, we show that the continued fractions for every rational num-
ber obtained by our algorithms always terminate. We conclude with several
remarks in Section 10.
2 p-adic continued fraction expansions
In what follows, α denotes an element of Q unless otherwise mentioned. We
p
mean by the p-adic expansion of α the series
∞
α= e pi (e =e (α)∈{0,1,...,p−1})
i i i
i=−∞
X
with e 6= 0 at most finitely many i ≤ 0. We define the integral and fractional
i
parts of α, denoted by [α] and hαi respectively, as
0 ∞
[α]:= e pi and hαi:= e pi. (2)
i i
i=−∞ i=1
X X
In this section, we consider expanding α into a continued fraction of the form
t1pk1
d + (k ∈Z , t ,d ∈Z \pZ (i≥1))
0 t2pk2 i >0 i i p p
d +
1 t3pk3
d +
2
...
with d ∈ Q such that d = [d ]. Note that this class of continued fractions
0 p 0 0
contains ones of the form (1).
3
Let t be a map from pZ \{0} to Z \pZ . Then, t(x)pvp(x) ∈ Z \pZ for
p p p x p p
all x ∈ pZ \{0}, where v (α) denotes the p-adic additive valuation of α. We
p p
consider a family of the maps of the form
T :pZ \{0}→pZ ,
p p
t(x)pvp(x) (3)
T(x):= −d(x),
x
where dis a mapfrompZ \{0}to Z \pZ . Since d(x)= t(x)pvp(x) −T(x)and
p p p x
T(x)∈pZ , we have [d(x)]= t(x)pvp(x) ∈{1,...,p−1} for all x∈pZ \{0}.
p x p
Hence, d is uniquely determinhed if the iimage of d, denoted by Im(d), satisfies
Im(d)⊂{1,...,p−1}.
Since
t(x)pvp(x)
x= ,
d(x)+T(x)
we have
t(Tn−1(hαi))pvp(Tn−1(hαi))
Tn−1(hαi)= ,
d(Tn−1(hαi))+Tn(hαi)
provided that Tn−1(hαi)6=0 (n∈Z ). Setting
>0
t =t Ti−1(hαi) ,
i
ki =vp(cid:0) Ti−1(hαi(cid:1)) ,
di =d (cid:0)Ti−1(hαi) (cid:1),
(cid:0) (cid:1)
for i∈{1,...,n}, we have
t1pk1
α=[α]+ .
t2pk2
d +
1 t3pk3
d +
2
... + tn−1pkn−1
tnpkn
d +
n−1 d +Tn(hαi)
n
Related to the continued fraction expansion of α, there occur three cases:
(i) hαi=0.
We do not expand hαi=0, and we have α=[α].
(ii) There exists N ∈ Z such that TN(hαi) = 0 and Tn(hαi) 6= 0 for all
>0
0≤n<N.
4
We can expand α into the finite continued fraction
t1pk1
α=[α]+ . (4)
t2pk2
d +
1 t3pk3
d +
2
... + tNpkN
d
N
(iii) Tn(hαi)6=0 for all n∈Z .
≥0
We can expand α into the infinite continued fraction
t1pk1
[α]+ . (5)
t2pk2
d +
1 t3pk3
d +
2
...
We will show that the continued fraction (5) converges to α in the suc-
ceeding section.
Remark 2.1.
(i) WecanconsideravarietyofmapsT. Infact,wewillgivethreealgorithms
of continued fraction expansion in Section 5; consequently, the expression
in continued fractions (4) and (5) are not uniquely determined for a given
α∈Q .
p
(ii) The continued fractions considered by Schneider [9] are generated by set-
ting t(x)≡1 and Im(d)⊂{1,...,p−1}.
3 Convergence of continued fractions
In this section, we show that the continued fraction described in Section 2
always converges to α for α ∈ Q . Without loss of generality, we may assume
p
that α∈pZ \{0} in this section.
p
We definetwosequences{p } and{q } intermsoft , k , d in(5)
n n≥−1 n n≥−1 i i i
by the following recursion formulas:
p−1 =1, p0 =0, pn =dnpn−1+tnpknpn−2 (n≥1),
(q−1 =0, q0 =1, qn =dnqn−1+tnpknqn−2 (n≥1).
Inthe caseofthe finite expansion(4),we define p andq forn with−1≤n≤
n n
N.
Lemmas 3.1–3.3 given below are easily seen (cf. [7]).
5
Lemma 3.1.
t1pk1 pn
= (n≥1).
t2pk2 qn
d +
1 t3pk3
d +
2
.. tnpkn
. +
d
n
Lemma 3.2.
p +Tn(α)p
n n−1
α= (n≥1).
q +Tn(α)q
n n−1
Lemma 3.3.
n
pn−1qn−pnqn−1 = −tipki (n≥1).
i=1
Y(cid:0) (cid:1)
We denoteby |α|p the p-adicabsolutevalue ofα∈Qp, i.e., |α|p :=1/pvp(α).
Lemma 3.4.
|q | =1 (n≥0).
n p
Proof. The claimistrue forn=0 andn=1. Assumingthat|q | =1holds for
i p
0 ≤ i ≤ n with n ≥ 1, we have |qn+1|p = |dn+1qn+tn+1pkn+1qn−1|p = 1 since
|dn+1qn|p =1 and |tn+1pkn+1qn−1|p ≤1/p.
Theorem 3.5.
(i) Let n be an integer with n≥1 or an integer with 1≤n≤N if there exists
an integer N ≥1 such that TN(α)=0. Then,
p |Tn(α)|
n p
α− =
(cid:12) qn(cid:12)p pPni=1ki
(cid:12) (cid:12)
(cid:12) (cid:12)
holds. In particular, (cid:12) (cid:12)
p 1
n
α− =
(cid:12) qn(cid:12)p pPni=+11ki
(cid:12) (cid:12)
holds if Tn(α)6=0. (cid:12) (cid:12)
(cid:12) (cid:12)
(ii) Let Tn(α)6=0 for all n≥1. Then,
p
n
lim =α
n→∞ qn
holds.
Proof. (i) By Lemma 3.2, we have
p p +Tn(α)p p Tn(α)(p q −p q )
n n n−1 n n−1 n n n−1
α− = − = .
q q +Tn(α)q q (q +Tn(α)q )q
n n n−1 n n n−1 n
6
By Lemma 3.3, we have
n |Tn(α)|
|Tn(α)(pn−1qn−pnqn−1)|p =(cid:12)Tn(α) −tipki (cid:12) = pPni=1kip.
(cid:12)(cid:12) iY=1(cid:0) (cid:1)(cid:12)(cid:12)p
(cid:12) (cid:12)
By Lemma 3.4, we have (cid:12) (cid:12)
|(q +Tn(α)q )q | =1.
n n−1 n p
Hence, we get
p |Tn(α)|
n p
α− = .
(cid:12) qn(cid:12)p pPni=1ki
(cid:12) (cid:12)
If Tn(α)6=0, then |Tn(α)|p(cid:12)(cid:12)=1/pkn(cid:12)(cid:12)+1, which implies
p 1
n
α− = .
(cid:12) qn(cid:12)p pPni=+11ki
(cid:12) (cid:12)
(cid:12) (cid:12)
The assertion (ii) immediate(cid:12)ly follow(cid:12)s from (i).
4 Two basic maps: T and T
1 2
We later propose three continued fraction algorithms, each of which gives an
eventually periodic expansion for every quadratic element of Q over Q and
p
givesa finite expansionforeveryrationalnumber. In this section, weintroduce
maps T and T on the basis of which we construct the algorithms.
1 2
We denote by A the set of all the elements of pZ which are algebraic over
p p
Q of degree at most two. For simplicity, we will abbreviate “algebraic over
Q” to “algebraic”, and “quadratic over Q” to “quadratic”. We mean, by the
minimal polynomial of an algebraic element α, the integral polynomial of the
lowest degree which has α as a root, whose leading coefficient is positive, and
whose coefficients are coprime. We denote the minimal polynomial of x ∈ A
p
by aX2+bX+c ifx is quadratic. We denote itby bX+c if x is rational. Note
that c 6= 0 if and only if x 6= 0. Let us define a map u : A \{0} → Z\pZ by
p
assigning
u(x):=c|c| ∈Z\pZ
p
to each x∈A \{0}. We define two maps T and T from A \{0} to A by
p 1 2 p p
u(x)pvp(x)
T (x):= −d (x),
1 1
x
and
−u(x)pvp(x)
T (x):= −d (x),
2 2
x
7
where d and d are maps from A \{0} to {1,...,p−1} which are uniquely
1 2 p
defined so as to let T (x) and T (x) belong to pZ , and thus to A , for every
1 2 p p
x∈ A \{0}. It is clear that T and T map any quadratic element of A to a
p 1 2 p
quadratic one. We remark that T and T belong to the family of the maps (3)
1 2
if we ignore their domains.
Ouralgorithmsintroducedinthenextsectionreduceexpansionsofalgebraic
elements of Q of degree at most two to expansions of those of pZ whose
p p
existence is guaranteedby the following well-known lemma.
Lemma4.1(Hensel’sLemma). Letf(X):=Xn+a Xn−1+···+a X+a ∈
n−1 1 0
Z[X], where n ∈ Z , a ∈ Z\pZ, and a ∈ pZ. Then, there exists unique
>0 1 0
α∈pZ such that f(α)=0.
p
In what follows, we call an element of pZ a quadratic Hensel root if it is
p
a root of X2+bX +c ∈ Z[X] where b ∈ Z\pZ, c ∈ pZ, and X2+bX +c is
irreducible. Likewise, we call an element of pZ a rational Hensel root if it is a
p
root of X +c∈Z[X] with c∈pZ (obviously, the root is −c).
Lemma 4.2. T and T map every quadratic Hensel root to a quadratic Hensel
1 2
root.
Proof. Let α∈pZ be an arbitraryquadratic Hensel root. By definition, α has
p
a minimal polynomial of the form X2 +bX +c ∈ Z[X] with b ∈ Z\pZ and
c ∈ pZ. We see that the conjugate ασ 6= α of α satisfies ασ ∈ Z \pZ since
p p
ασ = −b−α. Since α = c/ασ, we have |α| = |c| . Recalling the definition of
p p
u, we see that
u(α)pvp(α) c
= =ασ.
α α
Let
∞
ασ = e pi (e ∈{0,1,...,p−1}, e 6=0).
i i 0
i=0
X
Since ασ is a root of X2+bX +c, we have e (e +b) ≡ 0 (mod p). Let r be
0 0
an element of {1,...,p−1} satisfying r ≡ b (mod p). Since e 6= 0, we have
0
e +b≡0 (mod p), whichimplies e =p−r. Thus, d (α)=p−r,andwehave
0 0 1
T (α)=ασ−(p−r)∈pZ . By substituting X+(p−r) for X in X2+bX+c,
1 p
we have the minimal polynomial of T (α) given by
1
X2+{b+2(p−r)}X +(p−r)b+c+(p−r)2 ∈Z[X]. (6)
Since Z\pZ∋b+2(p−r)≡−r (mod p) and (p−r)b+c+(p−r)2 ∈pZ, we
see that T (α) is a quadratic Hensel root.
1
By a similar argument, we can show that T (α) = −ασ −r ∈ pZ and its
2 p
minimal polynomial is given by
X2+(−b+2r)X −rb+c+r2 ∈Z[X]. (7)
Since −b+2r ∈ Z\pZ and −rb+c+r2 ∈ pZ, we see that T (α) is also a
2
quadratic Hensel root.
8
Remark 4.1. Both maps T and T preservediscriminants of the minimal poly-
1 2
nomialsofquadraticHenselroots,i.e.,thediscriminantsof (6)and(7)areequal
to b2−4c.
5 Continued fraction algorithms
On the basis of T and T introduced in the previous section, we can consider
1 2
a variety of continued fraction algorithms which yield an eventually periodic
expansion for every quadratic element of Q and yield a finite expansion for
p
every rational number. In the present paper, we deal with three particular
algorithms. As in Section 4, the minimal polynomial of x ∈ A is denoted by
p
aX2+bX+c∈Z[X]forquadraticx, andby bX+c∈Z[X]forrationalx. Our
algorithms decide which map, T or T , is applied to a given x ∈ A \{0} on
1 2 p
the basis of two coefficients of its minimal polynomial, namely the coefficient b
ofX andtheconstanttermc,regardlessofthedegreeofx. Inthe following,we
specify our algorithms by specifying the map T : A \{0} → A used by each
p p
algorithm:
Algorithm A:
T(x):=T (x).
2
Algorithm B:
T (x) if b≥0,
2
T(x):=
(T1(x) if b<0.
Algorithm C:
T (x) if b≥0 and c>0,
2
T(x):=
(T1(x) otherwise.
6 Expansions of quadratic Hensel roots
In this section, we deal with the expansions of quadratic Hensel roots, on the
basis of which we expand general quadratic elements of Q . We show that
p
eachof our algorithmsgivesaneventually periodic expansionfor any quadratic
Hensel root. We will deal with the expansions of general quadratic elements of
Q and those of rational numbers in the subsequent sections.
p
WhenconsideringanexpansionofaquadraticHenselrootα,itisconvenient
to identify α with the pair (b,c) of coefficients of its minimal polynomial X2+
9
bX +c. In the following, we will do so and allow writing α = (b,c). Similarly,
we write T(α) also as T(b,c). We note that in view of (6) and (7), we have
T (b,c)= b+2(p−r),(p−r)b+c+(p−r)2 , (8)
1
(cid:0) (cid:1)
and
T (b,c)= −b+2r,−rb+c+r2 , (9)
2
(cid:0) (cid:1)
where r is an element of {1,...,p−1} satisfying r≡b (mod p).
Let S be the set of all quadratic Hensel roots, i.e.,
S = (b,c)∈Z2 | b∈Z\pZ, c∈pZ, and X2+bX+c is irreducible .
(cid:8) (cid:9)
We put
S :={(b,c)∈S | b>0, c>0},
1
S :={(b,c)∈S | b<0, c>0},
2
S :={(b,c)∈S | b<0, c<0},
3
S :={(b,c)∈S | b>0, c<0}.
4
We further put
R:={(b,c)∈S | 1≤b≤p−1},
R :={(b,c)∈S | 1≤b≤p−1},
1 1
R :={(b,c)∈S | 1≤b≤p−1}.
4 4
In the following subsections, we give Theorems 6.1, 6.2, and 6.4 which state
the periodicity of the continued fraction expansion obtained by Algorithms A,
B, and C, for any given quadratic Hensel root.
6.1 Expansions of quadratic Hensel roots by Algorithm A
Theorem 6.1. The expansion of every quadratic Hensel root obtained by Algo-
rithm A (i.e., T ) is purely periodic with period one or two.
2
Proof. Let (b,c) ∈ S. Let r be an element of {1,...,p−1} satisfying r ≡ b
(mod p). Then,T (b,c)= −b+2r,−rb+c+r2 . Using−b+2r≡r (mod p),
2
we easily see that T2(b,c) = (b,c). Thus, (b,c) is a purely periodic point with
2 (cid:0) (cid:1)
period two or one.
Remark 6.1. It is easy to see that (b,c)∈S is a fixed point of T if and only if
2
(b,c)∈R.
10