Algebra &
Number
Theory
Volume 8
2014
No. 2
Algebraicity of the zeta function associated to a
matrix over a free group algebra
Christian Kassel and Christophe Reutenauer
msp
, ALGEBRA AND NUMBER THEORY 8:2 (2014)
msp
dx.doi.org/10.2140/ant.2014.8.497
Algebraicity of the zeta function associated
to a matrix over a free group algebra
Christian Kassel and Christophe Reutenauer
Following and generalizing a construction by Kontsevich, we associate a zeta
function to any matrix with entries in a ring of noncommutative Laurent polyno-
mials with integer coefficients. We show that such a zeta function is an algebraic
function.
1. Introduction
Fix a commutative ring K . Let F be a free group on a finite number of generators
X 1 , . . . , X n and
KF = K hX 1 , X 1−1 , . . . , X n , X n−1 i
be the corresponding group algebra: equivalently, it is the algebra of noncommu-
tative Laurent polynomials with coefficients in K . Any element a ∈ KF can be
uniquely written as a finite sum of the form
X
a= (a, g)g,
g∈F
where (a, g) ∈ K .
Let M be a d ×d matrix with coefficients in KF. For any n ≥ 1, we may consider
the n-th power M n of M and its trace Tr(M n ), which is an element of KF. We
define the integer an (M) as the coefficient of 1 in the trace of M n :
an (M) = (Tr(M n ), 1). (1-1)
Let g M and PM be the formal power series
X
X tn
gM = an (M)t n
and PM = exp an (M) . (1-2)
n
n≥1 n≥1
They are related by
d log(PM )
gM = t .
dt
MSC2010: primary 05A15, 68Q70, 68R15; secondary 05E15, 14H05, 14G10.
Keywords: noncommutative formal power series, language, zeta function, algebraic function.
497
, 498 Christian Kassel and Christophe Reutenauer
We call PM the zeta function of the matrix M by analogy with the zeta function
of a noncommutative formal power series (see next section); the two concepts will
be related in Proposition 4.1.
The motivation for the definition of PM comes from the well-known identity
expressing the inverse of the reciprocal polynomial of the characteristic polynomial
of a matrix M with entries in a commutative ring
X n
1 n t
= exp Tr(M ) .
det(1 − t M) n
n≥1
Note that, for any scalar λ ∈ K , the corresponding series for the matrix λM
become
gλM (t) = g M (λt) and PλM (t) = PM (λt). (1-3)
Our main result is the following; it was inspired by Theorem 1 of [Kontsevich
2011]:
Theorem 1.1. For each matrix M ∈ Md (KF) where K = Q is the ring of rational
numbers, the formal power series PM is algebraic.
The special case d = 1 is due to Kontsevich [2011]. A combinatorial proof in
the case d = 1 and F is a free group on one generator appears in [Reutenauer and
Robado 2012].
Observe that by the rescaling equalities (1-3) it suffices to prove the theorem
when K = Z is the ring of integers.
It is crucial for the veracity of Theorem 1.1 that the variables do not commute:
for instance, if a = x + y + x −1 + y −1 ∈ Z[x, x −1 , y, y −1 ], where x and y are
commuting variables, then exp( n≥1 (a n , 1)t n /n) is a formal power series with
P
integer coefficients but not an algebraic function (this follows from Example 3 in
[Bousquet-Mélou 2005, §1]).
The paper is organized as follows. In Section 2, we define the zeta function ζ S
of a noncommutative formal power series S and show that it can be expanded as
an infinite product under a cyclicity condition that is satisfied by the characteristic
series of cyclic languages.
In Section 3, we recall the notion of algebraic noncommutative formal power
series and some of their properties.
In Section 4, we reformulate the zeta function of a matrix as the zeta function of
a noncommutative formal power series before giving the proof of Theorem 1.1; the
latter follows the steps sketched in [Kontsevich 2011] and relies on the results of the
previous sections as well as on an algebraicity result by André [2004] elaborating
on an idea of D. and G. Chudnovsky.
We concentrate on two specific matrices in Section 5. We give a closed formula
for the zeta function of the first matrix; its nonzero coefficients count the planar
Number
Theory
Volume 8
2014
No. 2
Algebraicity of the zeta function associated to a
matrix over a free group algebra
Christian Kassel and Christophe Reutenauer
msp
, ALGEBRA AND NUMBER THEORY 8:2 (2014)
msp
dx.doi.org/10.2140/ant.2014.8.497
Algebraicity of the zeta function associated
to a matrix over a free group algebra
Christian Kassel and Christophe Reutenauer
Following and generalizing a construction by Kontsevich, we associate a zeta
function to any matrix with entries in a ring of noncommutative Laurent polyno-
mials with integer coefficients. We show that such a zeta function is an algebraic
function.
1. Introduction
Fix a commutative ring K . Let F be a free group on a finite number of generators
X 1 , . . . , X n and
KF = K hX 1 , X 1−1 , . . . , X n , X n−1 i
be the corresponding group algebra: equivalently, it is the algebra of noncommu-
tative Laurent polynomials with coefficients in K . Any element a ∈ KF can be
uniquely written as a finite sum of the form
X
a= (a, g)g,
g∈F
where (a, g) ∈ K .
Let M be a d ×d matrix with coefficients in KF. For any n ≥ 1, we may consider
the n-th power M n of M and its trace Tr(M n ), which is an element of KF. We
define the integer an (M) as the coefficient of 1 in the trace of M n :
an (M) = (Tr(M n ), 1). (1-1)
Let g M and PM be the formal power series
X
X tn
gM = an (M)t n
and PM = exp an (M) . (1-2)
n
n≥1 n≥1
They are related by
d log(PM )
gM = t .
dt
MSC2010: primary 05A15, 68Q70, 68R15; secondary 05E15, 14H05, 14G10.
Keywords: noncommutative formal power series, language, zeta function, algebraic function.
497
, 498 Christian Kassel and Christophe Reutenauer
We call PM the zeta function of the matrix M by analogy with the zeta function
of a noncommutative formal power series (see next section); the two concepts will
be related in Proposition 4.1.
The motivation for the definition of PM comes from the well-known identity
expressing the inverse of the reciprocal polynomial of the characteristic polynomial
of a matrix M with entries in a commutative ring
X n
1 n t
= exp Tr(M ) .
det(1 − t M) n
n≥1
Note that, for any scalar λ ∈ K , the corresponding series for the matrix λM
become
gλM (t) = g M (λt) and PλM (t) = PM (λt). (1-3)
Our main result is the following; it was inspired by Theorem 1 of [Kontsevich
2011]:
Theorem 1.1. For each matrix M ∈ Md (KF) where K = Q is the ring of rational
numbers, the formal power series PM is algebraic.
The special case d = 1 is due to Kontsevich [2011]. A combinatorial proof in
the case d = 1 and F is a free group on one generator appears in [Reutenauer and
Robado 2012].
Observe that by the rescaling equalities (1-3) it suffices to prove the theorem
when K = Z is the ring of integers.
It is crucial for the veracity of Theorem 1.1 that the variables do not commute:
for instance, if a = x + y + x −1 + y −1 ∈ Z[x, x −1 , y, y −1 ], where x and y are
commuting variables, then exp( n≥1 (a n , 1)t n /n) is a formal power series with
P
integer coefficients but not an algebraic function (this follows from Example 3 in
[Bousquet-Mélou 2005, §1]).
The paper is organized as follows. In Section 2, we define the zeta function ζ S
of a noncommutative formal power series S and show that it can be expanded as
an infinite product under a cyclicity condition that is satisfied by the characteristic
series of cyclic languages.
In Section 3, we recall the notion of algebraic noncommutative formal power
series and some of their properties.
In Section 4, we reformulate the zeta function of a matrix as the zeta function of
a noncommutative formal power series before giving the proof of Theorem 1.1; the
latter follows the steps sketched in [Kontsevich 2011] and relies on the results of the
previous sections as well as on an algebraicity result by André [2004] elaborating
on an idea of D. and G. Chudnovsky.
We concentrate on two specific matrices in Section 5. We give a closed formula
for the zeta function of the first matrix; its nonzero coefficients count the planar
