The probability of rectangular unimodular matrices over $\F_q[x]$

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by  Xiangqian Guo

THE PROBABILITY OF RECTANGULAR UNIMODULAR
MATRICES OVER F [x]
q
XIANGQIANGUOANDGUANGYUYANG
Abstract. In this note, we compute the probability that a k×n matrix
can be extended to an n×n invertible matrix over Fq[x], which turns out
1 to be (1−qk−n)(1−qk−1−n)···(1−q1−n). Connections with Dirichlet’s
1 density theorem on the co-prime integers and its various generalizations are
0 alsopresented.
2

l
u
J
1. introduction and main results
8
1
For any set A of positive integers, its natural density is defined as

] |A∩{1,2,...,N}|
R (1) D(A):= lim ,
N→∞ N
P
. providedthelimitexists,where|·|denotesthecardinalityofthecorrespondingset.
h
Dirichlet [2] discovered an interesting density theorem that asserts the probability
t
a of two integers to be co-prime is 6/π2, that is,
m
|{(m,n)∈N2 : 16m,n6N,gcd(m,n)=1}| 6
[ (2) lim =ζ−1(2)= ,
N→∞ N2 π2
1
where gcd(m,n) denotes the greatest common divisor of m and n, and ζ(s) is the
v
7 Riemann’s zeta function. Moreover,the probability of n integers to be co-prime is
6
(3)
3
3 lim |{(m1,...,mn)∈Nn : 16m1,...,mn 6N,gcd(m1,...,mn)=1}| =ζ−1(n).
7. N→∞ Nn
0 Kubota and Sugita [5] gave a rigorous probabilistic interpretation to Dirichlet’s
1 densitytheorem. Forthedeeplinksbetweenprobabilitytheoryandnumbertheory,
1
please refer to Tenenbaum [12], Kubilius [4] and Kac [3].
:
v Denote by M (R) the set of all k × n matrices over a ring R. A matrix
k×n
i
X A ∈ Mk×n(R) is called unimodular if it can be extended to an n×n invertible
matrix. Using the conceptof unimodular, we can give a re-statementof Dirichlet’s
r
a densitytheoremanditsgeneralization: the probabilitythatarandom1×ninteger
matrix is unimodular is ζ−1(n). Naturally, we can consider the matrix form of the
Dirichlet’sdensitytheorem: theprobabilitythatarandomk×n(16k 6n)integer
matrix is unimodular? The key is to define the analogous natural density for the
sets of the integer matrices. Maze, Rosenthal and Wagner [7] studied this problem
and obtained that the probability that a random k ×n matrix is unimodular is
n ζ−1(j). Furthermore, [7] also gave some interesting historical remarks
Qj=n−k+1
for Dirichlet’s density theorem.
Then it is natural to consider the same question for matrices over polynomial
rings of fields. As remarked in [7], the concept of natural density does not extend
2000 Mathematics Subject Classification. 11C99,15B33,60B15.
Keywordsandphrases. Naturaldensity,unimodularmatrices,Riemann’szetafunction,q-zeta
function,finitefield,polynomialring.
1
2 X.GUOANDG.YANG
naturally to the ring F[x] for a general field F. Fortunately, this can be done for
the finite fields.
LetF beafinitefieldconsistingofq elements,andF [x]bethepolynomialring
q q
over F . To enumerate F [x], let Σ be the set of all vectors α = (a ,a ,···) with
q q 0 1
a ∈{0,1,··· ,q−1} and a =0 for sufficiently large i. Then there is a one-to-one
i i
map χ:Σ→Z =N∪{0} defined by χ(a ,a ,···)= ∞ a qi. For convenience,
we denote α =+χ−1(m) and m =χ(α). 0The1n for allPmi=∈0Zi , we set
m α +
∞
f (x):= a xi, with α =(a ,a ,···).
m X i m 0 1
i=0
By a rigorous probabilistic method, Sugita and Takanobu [10] determined the
probability of two polynomials over F to be co-prime when q is a prime, that is,
q
|{(m,n)∈{0,1,...,N −1}2 : gcd(f ,f )=1}| 1
m n
(4) lim =1− .
N→∞ N2 q
Recently,severalauthorsconsideredthequestionoftheprobabilityofnpolynomials
to be co-prime in F [x]. Morrison[8] introduced a concept of natural density
q
|S∩F |
(5) D(S):= lim d , S ⊆F [x],
q
e d→∞ |Fd|
where F is the set of all the polynomials in F[x] with degree no more than d. He
d
first treated this problem in a heuristic way. Then through an elegant as well as
rigorous argument, he deduced the following formula,
(6)
|{(f1,...,fn)∈Fnq[x]:gcd(f1,...,fn)=1,deg(fi)6d,16i6n}| 1
lim =1− .
d→∞ |{f ∈Fq[x]: deg(f)6d}|n qn−1
BenjaminandBennett[1]alsostudiedthis problem;their methodsaremuchinter-
esting using the Euclidean algorithm which is one of the oldest algorithms. Essen-
tially, they also got the same conclusion.
Our object in this paper is to extend the main results in [7] to the polynomial
ringF [x]foranyprimepowerq,thatis,thematrixformoftheMorrison’stheorem
q
(6). Denote F=F andF[x]=F [x] for short. Let M:=M (F[x]) be the set of
q q k×n
all k×n matrices over F[x] and M be the subset of M consisting of all matrices
N
with entries in {f (x),f (x),··· ,f (x)}. For any subset S ⊆ M, we define the
0 1 N
natural density of S in M to be
|S∩M |
N
(7) D(S):= lim ,
N→∞ |MN|
if the above limit (7) exists. Note that, in the special case k =1, the limit in (5) is
just the limit of a subsequence in (7) with N = qd. Thus our definition of natural
density is a more rigorous one.
Todescribeourresult,i.e.,todeterminethenaturaldensityofk×nunimodular
matrices over F[x], we introduce the following q-zeta function
∞
1 1
(8) ζ (j):= (1− )−1 = (1− )−ϕm,
q Y qjdeg(f) Y qjm
f m=1
where f goes through all irreducible polynomials in F[x] and ϕ is the number of
m
irreducible polynomials in F[x] with degree m. In the above statements and what
follows, by irreducible polynomials we always mean monic irreducible
polynomials (the constant polynomial 1 is excluded as usual). Nowweare
in the position to give our main result.
THE PROBABILITY OF RECTANGULAR UNIMODULAR MATRICES OVER Fq[x] 3
Theorem 1. Let E be the set of all k×n unimodular matrices over F[x], then the
natural density of E is
∞ n
1 1
(9) D(E)= (1− )= (1− )ϕm = ζ−1(j).
Y qjdeg(f) Y qjm Y q
f m=1 j=n−k+1
Remark 2. In other words, the probability that a k ×n polynomial matrix can
be completed by an (n−k)×n matrix into an n×n invertible matrix over F[x]
is n ζ−1(j). And via introducing the q-zeta function ζ (s), we obtain a
Qj=n−k+1 q q
similar formula to the one in [7] for integer matrices
n
D(E)= ζ−1(j),
Y
e j=n−k+1
where E denotes the set of all k×n unimodular matrices over Z.
e
Remark 3. There is an interesting equation
∞
1
(10) (1−tdeg(f))−1 = qltl = ,
Y X 1−qt
f l=0
where f is over all irreducible polynomials. For an interpretation of this equation,
see [8]. Putting t=q−j,j ∈N in (10), we get
1
(11) ζ−1(j)=1− .
q qj−1
Hence, we obtain an explicit formula for the natural density of E,
1 1 1
(12) D(E)=(1− )(1− )···(1− ),
qn−k qn−k+1 qn−1
and an interesting identity concerning {ϕ },
m
∞
1 1
(13) (1− )ϕm =1− , ∀j ∈N.
Y qjm qj−1
m=1
Remark 4. In particular, if we put k = 1, Theorem 1 shows that the probability
that n polynomials over the finite field F to be co-prime is
∞
1 1
ζ−1(n)= (1− )ϕm =1− .
q Y qnm qn−1
m=1
This extends Dirichlet’s density theorem (2) and yields the same conclusion as
Morrison[8]andBenjaminandBennett[1]. However,noticingthatwehavedefined
theconcept ofnaturaldensityinadifferentway fromthem, as wementionedbefore,
one may find that our result actually is stronger than theirs.
We end this section with some remarks on the square case k = n. One may
notice that our proof of Theorem 1 in §2 does not apply to this case (also, ζ (1) is
q
not well defined in this case). However, we have the following result,
Proposition 5. The natural density of n×n unimodular matrices over F [x] is 0.
q
Remark 6. The proof of Proposition 5 is quite simple and similar to the proof of
Lemma 5 in [7], so we omit the details. If we make the convention that ζ−1(1)=0,
q
Theorem 1 also holds for k =n.
4 X.GUOANDG.YANG
2. Proof of Main result
In this section, we shall give the proof of Theorem 1. Before this, we need some
preparations.
Let P be a finite set of irreducible polynomials in F[x] and P be the set of all
irreducible polynomials in F[x]. Denote by E the set of all mbatrices A ∈ M =
P
M (F[x]) such that the gcd of all full rank minors (i.e., minors of rank k) is
k×n
co-prime to all elements in P. Recall that E is the set of all unimodular matrices
inM,thatis,eachmatrixinE canbeextendedtoann×ninvertiblematrix. Itis
well knownthat A∈M is unimodular if and only if the gcd of all full rank minors
is 1 (See [9], [11] and [13] for more details). Thus we have E = E .
TP P
Lemma 7. Let E be defined as above, then we have
P
k−1 n
(14) D(E )= (1−q(j−n)deg(f))= (1−q−jdeg(f)).
P Y Y Y Y
j=0f∈P j=n−k+1f∈P
Proof. Denote f := f and d = deg(f ). For any positive integer N,
P Qf∈P P P
consider the maps,
(15) π : M −→M (F[x]/(f ))
N k×n P
and
(16)
ψ : M (F[x]/(f ))−→M  F[x]/(f)−→ M (F[x]/(f)),
k×n P k×n Y Y k×n
f∈P  f∈P
where (f ) and (f) denote the ideals generated by f and f respectively. Note
P P
that π is the canonical projection via modulo f and ψ is a composition of two
P
canonical isomorphisms as F vector spaces.
Taken any A ∈ M . It is easy to see that A ∈ E if and only if the com-
N P
ponent of ψ ◦π(A) in M (F[x]/(f)) is full rank for each f ∈ P. Since f ∈ P
k×n
is irreducible, we know that F[x]/(f) ≃ F . Let F denote the set of
qdeg(f) qdeg(f)
all full rank k × n matrices over the finite field F . It is well known that
qdeg(f)
|F |= k−1(qndeg(f)−qjdeg(f)) (See [6] page 455, for example).
qdeg(f) Qj=0
First suppose that N =mqdP −1 for some m∈N. Then it is easy to see
{f (x): 06l 6N}={f (x)xdP +f (x): 06s6m−1,06t6qdP −1}.
l s t
For any fixed 06s6m−1, the following projection is one-to-one:
{f (x)xdP +f (x): 06t6qdP −1}−→F[x]/(f )
s t P
and the canonical projection
{f (x): 06l 6N}−→F[x]/(f )
l P
is m-to-one. As a result, the projection map π in (15) is mkn-to-one. Thus we
obtain that
|E ∩M |=mkn· |F |
P N Y qdeg(f)
f∈P
k−1
=mkn· qndeg(f)(1−q(j−n)deg(f))
Y Y
f∈Pj=0
k−1
(17) =(mqdP)kn· (1−q(j−n)deg(f))
Y Y
f∈Pj=0
THE PROBABILITY OF RECTANGULAR UNIMODULAR MATRICES OVER Fq[x] 5
Now suppose that N is any positive integer. There exists some m,r ∈ Z such
+
that N +1 = mqdP +r with 0 6 r < qdP. For convenience, set N := mqdP −1.
Then by the definition of the natural density, we have e
|E ∩M |
P N
D(E )= lim
P
N→∞ |MN|
(18) = lim |EP ∩MNe|+|EP ∩(MN −MNe)|.
N→∞ (N +1)kn
Note that |MN −MNe|6rkn(N +1)kn−1, which gives that
(19) lim |EP ∩(MN −MNe)| =0.
N→∞ (N +1)kn
Thus, from (18), (19) and (17), we obtain
D(E )= lim |EP ∩MNe|
P N→∞ (N +1)kn
(N +1−r)kn k−1(1−q(j−n)deg(f))
= lim Qf∈PQj=0
N→∞ (N +1)kn
k−1
= (1−q−(n−j)deg(f))
Y Y
f∈Pj=0
n
= (1−q−jdeg(f)).
Y Y
j=n−k+1f∈P
This completes the proof of Lemma 7. (cid:3)
Proof of Theorem 1. Suppose that k < n. For any irreducible polynomial
f ∈F [x], let H ⊆M be the set of all matrices whose gcd of all full rank minors
q f
is divisible by f. Denote q :=qdeg(f), then by Lemma 7,
f
D(H )=1−D(E )
f {f}
n
=1− (1−q−jdeg(f))
Y
j=n−k+1
n
<1−1− q−j
X f
 j=n−k+1 
∞
1 2
< q−j = 6 .
X f qn−k(q −1) q2
j=n−k+1 f f f
Let P be the set of all irreducible polynomials with degree no more than t and
t
denote E =E . Since
t Pt
(E \E)⊆ H ,
t [ f
b
f∈P\Pt
6 X.GUOANDG.YANG
we have
limsup|(Et\E)∩MN| 6limsup|(Sf∈Pb\PtHf)∩MN|
|M | |M |
N→∞ N N→∞ N
6limsupPf∈Pb\Pt|Hf ∩MN|
|M |
N→∞ N
|H ∩M |
6 limsup f N
X |M |
b N→∞ N
f∈P\Pt
2
= D(H )6
X f X q2
b b f
f∈P\Pt f∈P\Pt
∞
2 2ϕ
m
= = .
X q2deg(f) X q2m
f∈Pb\Pt m=t+1
Itis wellknownthatallirreduciblepolynomials withdegreem candivide xqm−x,
which has no multiple roots. Thus mϕ 6qm and
m
∞
|(E \E)∩M | 2
limsup t N 6
|M | X mqm
N→∞ N
m=t+1
∞
2 2
6 = .
X qm qt(q−1)
m=t+1
Nownote thatE∩M ⊆E ∩M andE∩M =E ∩M −(E \E)∩M ,
N t N N t N t N
which imply that
|E∩M | |E ∩M | |(E \E)∩M |
liminf N >liminf t N −limsup t N
N→∞ |MN| N→∞ |MN| N→∞ |MN|
2
>D(E )− ,
t qt(q−1)
and
|E∩M | |E ∩M | |(E \E)∩M |
limsup N 6limsup t N −liminf t N
N→∞ |MN| N→∞ |MN| N→∞ |MN|
6D(E ),
t
for all t ∈ N. Let t tend to ∞, and recall that the number of monic irreducible
polynomials with degree m in F[x] is ϕ . From Lemma 7 and the definition of
m
q-zeta function, we can conclude that
|E∩M |
N
lim = lim D(E )
t
N→∞ |MN| t→∞
n
= lim (1−q−jdeg(f))
t→∞ Y Y
j=n−k+1f∈Pt
n t
= lim (1−q−jm)ϕm
t→∞ Y Y
j=n−k+1m=1
n +∞ n
= (1−q−jm)ϕm = ζ−1(j).
Y Y Y q
j=n−k+1m=1 j=n−k+1
This completes the proof of Theorem 1. (cid:3)
THE PROBABILITY OF RECTANGULAR UNIMODULAR MATRICES OVER Fq[x] 7
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(X.Guo)DepartmentofMathematics,ZhengzhouUniversity,HenanProvince,450001,
China.
E-mail address: [email protected]
(G.Yang)DepartmentofMathematics,ZhengzhouUniversity,HenanProvince,450001,
China.
E-mail address: study [email protected]