There are many examples of several variable polynomials whose Mahler measure is expressed in terms of special values of polylogarithms. These examples are expected to be related to computations of regulators, as observed by Deninger [D] and, later, by Rodriguez-Villegas [R] and Maillot [M]. While Rodriguez-Villegas made this relationship explicit for the two-variable case, it is our goal to understand the three-variable case and shed some light on the examples with more variables. 1

Matilde N. Lalín

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An algebraic integration for Mahler Measure
Junior Number Theory Seminar – University of Texas at Austin
February 24th, 2004
Matilde N. Lal´ ın
1. Mahler Measure
Deﬁnition 1 For P ∈ C[x±1
1 ,...,x±1
n ], the (logarithmic) Mahler measure is deﬁned by
m(P) =
1
(2πi)n
Z
Tn
log|P(x1,...,xn)|
dx1
x1
...
dxn
xn
(1)
There is a simple expression for the Mahler measure in the one-variable case, as a
function on the roots of the polynomial. The question is, what happens with several
variables?
2. Polylogarithms
Deﬁnition 2 The kth polylogarithm is the function deﬁned by the power series
Lik(x) :=
∞ X
n=1
xn
nk x ∈ C, |x| < 1 (2)
This function can be continued analytically to C \ (1,∞).
In order to avoid discontinuities, and to extend this function to the whole complex plane,
several modiﬁcations have been proposed. Zagier [10] considers the following version:
Pk(x) := Rek


k X
j=0
2jBj
j!
(log|x|)jLik−j(x)

 (3)
where Bj is the jth Bernoulli number, Li0(x) ≡ −1
2 and Rek denotes Re or Im depending
on whether k is odd or even.
This function is one-valued, real analytic in P1(C)\{0,1,∞} and continuous in P1(C).
Moreover, Pk satisfy very clean functional equations. The simplest ones are
Pk
µ
1
x
¶
= (−1)k−1Pk(x) Pk(¯ x) = (−1)k−1Pk(x)
there are also lots of functional equations which depend on the index k. For instance, for
k = 2, we have the well-known ﬁve-term relation1
D(x) + D(1 − xy) + D(y) + D
µ
1 − y
1 − xy
¶
+ D
µ
1 − x
1 − xy
¶
= 0 (4)
Polylogarithms appear in many examples as the Mahler measures of polynomials in
several variables.
1Note that P2 = D, the Bloch–Wigner dilogarithm
1The most famous and basic examples in two and three variables are due to Smyth [9]:
m(1 + x + y) =
1
π
D(ζ6) =
3
√
3
4π
L(χ−3,2) (5)
m(1 + x + y + z) =
7
2π2ζ(3) (6)
Note that Pk(1) = Lik(1) = ζ(k).
3. The two-variable case
In order to understand the formulas for the two variables polynomials, Rodriguez-Villegas
[8] has proposed the following construction inspired in Deninger’s work. This was later
developed by Boyd and Rodriguez-Villegas [1], [2].
Given a smooth projective curve C and x, y rational functions (x,y ∈ C(C)∗), deﬁne
η(x,y) = log|x|dargy − log|y|dargx (7)
Here
dargx = Im
µ
dx
x
¶
(8)
is well deﬁned in C in spite of the fact that arg is not. η is a 1-form in C \ S, where S is
the set of zeros and poles of x and y. It is also closed, because of
dη = Im
µ
dx
x
∧
dy
y
¶
= 0
Let P ∈ C[x,y]. Write
P(x,y) = ad(x)yd + ... + a0(x)
P(x,y) = ad(x)
d Y
n=1
(y − αn(x))
Then by Jensen’s formula,
m(P) = m(ad) +
1
2πi
d X
n=1
Z
T1
log+ |αn(x)|
dx
x
= m(ad) −
1
2π
Z
γ
η(x,y) (9)
Here γ is the union of paths in C = {P(x,y) = 0} where |x| = 1 and |y| ≥ 1. Also note
that ∂γ = {(x,y) ∈ C2 ||x| = |y| = 1,P(x,y) = 0}
We want to arrive to one of these two situations:
1. η is exact, and ∂γ 6= 0. In this case we can integrate using Stokes Theorem.
2. η is not exact and ∂γ = 0. In this case we can compute the integral by using Residue’s
Theorem.
Here we will only care about the ﬁrst case. Under certain conditions (see [8]) η can be
extended to C and becomes a closed form there. We need it to be exact. In fact, we have:
2Theorem 3
η(x,1 − x) = dD(x) (10)
We will associate η with an element in H1(C \ S,R) in the following way. Given
[γ] ∈ H1(C \ S,Z),
[γ] →
Z
γ
η (11)
(we identify H1(C \ S,R) with H1(C \ S,Z)0).
Note the following
Theorem 4 η satisﬁes the following properties
1. η(x,y) = −η(y,x)
2. η(x1x2,y) = η(x1,y) + η(x2,y)
3. η(x,1 − x) = 0 in H1(C,R)
As a consequence, η is a symbol, and can be factored through K2(C(C)) (by Matsumoto’s
Theorem). Then we can guarantee that η(x,y) is exact by having {x,y} is trivial in
K2(C(C)) ⊗ Q. (Tensoring with Q kills roots of unity, which is ﬁne, since η is trivial on
them).
In general, if
x ∧ y =
X
j
rjzj ∧ (1 − zj)
in
V2(C(C)∗) ⊗ Q, then
η(x,y) = d


X
j
rjD(zj)

 = dD


X
j
rj[zj]


We have γ ⊂ C such that
∂γ =
X
k
²k[wk] ²k = ±1
where wk ∈ C(C), |x(wk)| = |y(wk)| = 1. Then
2πm(P) = D(ξ) for ξ =
X
k
X
j
rj[zj(wk)]
In order to interpret Smyth’s case, take P(x,y) = x + y + 1. Writing x = e2πiθ, the
path of integration becomes
γ(θ) = 1 + e2πiθ, θ ∈ [1/6;5/6] ⇒ ∂γ = [ξ6] − [¯ ξ6]
2πm(x + y + 1) = D(ζ6) − D(¯ ζ6) = 2D(ζ6)
4. The three-variable case
3We are going to extend this situation to three variables. We will take
η(x,y,z) = log|x|
µ
1
3
dlog|y|dlog|z| − dargy dargz
¶
+log|y|
µ
1
3
dlog|z|dlog|x| − dargz dargx
¶
+log|z|
µ
1
3
dlog|x|dlog|y| − dargxdargy
¶
Then η veriﬁes
dη(x,y,z) = Re
µ
dx
x
∧
dy
y
∧
dz
z
¶
We can express the Mahler measure of P
m(P) = −
1
(2π)2
Z
Γ
η(x,y,z)
(we are taking P monic to simplify notation).
Where Γ = {P(x,y,z) = 0} ∩ {|x| = |y| = 1,|z| ≥ 1}. We are integrating on a subset
of the surface {P(x,y,z) = 0}. The diﬀerential form is deﬁned in this surface.
Suppose we have
x ∧ y ∧ z =
X
rixi ∧ (1 − xi) ∧ yi
Since we have
η(x,1 − x,y) = dω(x,y)
where
ω(x,y) = −D(x)dargy +
1
3
log|y|(log|1 − x|dlog|x| − log|x|dlog|1 − x|)
Then
Z
Γ
η(x,y,z) =
X
ri
Z
Γ
η(xi,1 − xi,yi) =
X
ri
Z
∂Γ
ω(xi,yi)
Where ∂Γ = {P(x,y,z) = 0}∩{|x| = |y| = |z| = 1} = {P(x,y,z) = P(x−1,y−1,z−1) =
0}∩{|x| = |y| = 1}. Note that we are integrating now on a path {|x| = |y| = 1} inside the
curve {P(x,y,z) = P(x−1,y−1,z−1) = 0}. The diﬀerential form ω is deﬁned in this new
curve (this way of thinking the integral over a new curve has been proposed by Maillot).
Now it makes sense to try to apply Stokes again. The condition is the following:
Supposse we have
[x]2 ⊗ y =
X
ri[xi]2 ⊗ xi
Since we have
ω(x,x) = dP3(x)
Then we have as before:
Z
γ
ω(x,y) =
X
ri P3(xi)|∂γ
5. The K-theory context
4We follow Goncharov, [5], [6]. Given a ﬁeld F, we deﬁne subgroups Ri(F) ⊂ Z[P1
F] as
R1(F) := [x] + [y] − [xy]
R2(F) := [x] + [y] + [1 − xy] +
·
1 − x
1 − xy
¸
+
·
1 − y
1 − xy
¸
R3(F) := certain functional equation of the trilogarithm
Deﬁne
Bi(F) := Z[P1
F]/Ri(F) (12)
The idea is that Bi(F) is the place where Pi naturally acts. We have the following
complexes:
BF(3) ⊗ Q : B3(F)Q
δ3
1 −→ (B2(F) ⊗ F∗)Q
δ3
2 −→ (∧3F∗)Q
BF(2) ⊗ Q : B2(F)Q
δ2
1 −→ (∧2F∗)Q
BF(1) ⊗ Q : F∗
Q
(Bi(F) is placed in degree 1).
δ3
1([x]3) = [x]2 ⊗ x δ3
2([x]2 ⊗ x) = x ∧ (1 − x) ∧ y δ2
1([x]2) = x ∧ (1 − x)
Proposition 5
H1(BF(1)) ∼ = K1(F) (13)
H2(BF(2)) ∼ = K2(F) (14)
H3(BF(3)) ∼ = KM
3 (F) (15)
Goncharov [5] conjectures:
Hi(BF(3) ⊗ Q) ∼ = K
[3−i]
6−i (F)
Where K
[i]
n (F) is a certain quotient in a ﬁltration of Kn(F).
Note that our ﬁrst condition is that x ∧ y ∧ z = 0 in H3(BF(3) ⊗ Q) and the second
condition is [xi]2⊗yi = 0 in H2(BF(3)⊗Q). Hence, everything can be translated as certain
elements in diﬀerent K-theories must be zero, which is analogous to the two-variable case.
6. Example
The importance of the following example is that in spite of being quite hard to compute
the example by simple integrals, it can be solved with this algebraic method.
Consider
Res(x + ytm + tn+m,z + wtm + tn+m) = (z − x)n+m − (wx − yz)m(y − w)n
which reduces to compute the Mahler measure of:
z =
(1 − x)m(1 − y)n
(1 − xy)m+n
5η(x,y,z) = mη(x,y,1 − x) + nη(x,y,1 − y) − (m + n)η(x,y,1 − xy)
= −mη(x,1 − x,y) + nη(y,1 − y,x) + mη(xy,1 − xy,y) − nη(xy,1 − xy,x)
∆ = m([xy]2 ⊗ y − [x]2 ⊗ y) − n([xy]2 ⊗ x − [y]2 ⊗ x)
Now
[x]2 + [y]2 + [1 − xy]2 +
·
1 − x
1 − xy
¸
2
+
·
1 − y
1 − xy
¸
2
= 0
Let
x1 =
1 − x
1 − xy
y1 =
1 − y
1 − xy
ˆ x1 = 1 − x1 ˆ y1 = 1 − y1
then
∆ = m([y]2 ⊗ y + [x1]2 ⊗ y + [y1]2 ⊗ y) − n([x]2 ⊗ x + [x1]2 ⊗ x + [y1]2 ⊗ x)
Now note that x = ˆ x1
y1, y =
ˆ y1
x1.
= m([y]2 ⊗ y + [x1]2 ⊗ ˆ y1 − [x1]2 ⊗ x1 + [y1]2 ⊗ ˆ y1 − [y1]2 ⊗ x1)
−n([x]2 ⊗ x + [x1]2 ⊗ ˆ x1 − [x1]2 ⊗ y1 + [y1]2 ⊗ ˆ x1 − [y1]2 ⊗ y1)
= m([y]2 ⊗ y + [x1]2 ⊗ ˆ y1 − [x1]2 ⊗ x1 − [ˆ y1]2 ⊗ ˆ y1 − [y1]2 ⊗ x1)
−n([x]2 ⊗ x − [ˆ x1]2 ⊗ ˆ x1 − [x1]2 ⊗ y1 + [y1]2 ⊗ ˆ x1 − [y1]2 ⊗ y1)
Recall that
xm
1 yn
1 ˆ xn
1ˆ ym
1 = 1
in the curve.
Then
[x1]2 ⊗ yn
1 ˆ ym
1 − [y1]2 ⊗ xm
1 ˆ xn
1 = −[x1]2 ⊗ xm
1 ˆ xn
1 + [y1]2 ⊗ yn
1 ˆ ym
1
= −m[x1]2 ⊗ x1 + n[ˆ x1]2 ⊗ ˆ x1 + n[y1]2 ⊗ y1 − m[ˆ y1]2 ⊗ ˆ y1
Then
∆ = m([y]2 ⊗ y − [ˆ y1]2 ⊗ ˆ y1 − [x1]2 ⊗ x1 − [ˆ y1]2 ⊗ ˆ y1 − [x1]2 ⊗ x1)
−n([x]2 ⊗ x − [ˆ x1]2 ⊗ ˆ x1 − [y1]2 ⊗ y1 − [ˆ x1]2 ⊗ ˆ x1 − [y1]2 ⊗ y1)
∆ = m([y]2 ⊗ y − 2[ˆ y1]2 ⊗ ˆ y1 − 2[x1]2 ⊗ x1) − n([x]2 ⊗ x − 2[ˆ x1]2 ⊗ ˆ x1 − 2[y1]2 ⊗ y1)
We now need to check the path of integration.
Let us write x = e2iα, y = e2iβ, for −π
2 ≤ α,β ≤ π
2. Also, we can suppose 0 ≤ α ≤ π
2
and
m(P) = −
1
2π2
Z
− π
2 ≤β≤ π
2 ,0≤α≤ π
2
η(x,y,z)
6Let a = sinα
sin(α+β), b =
sinβ
sin(α+β). Then
x1 = ae−iβ y1 = be−iα ˆ x1 = beiα ˆ y1 = aeiβ
Now, a, b and be though as the sides of a triangle with angles α, β opposing a and b
respectively. See [4]. We know that we integrate in
0 ≤ b ≤ 1 1 − b ≤ a ≤ 1 + b
1 ≤ b b − 1 ≤ a ≤ 1 + b
And also
ambn = 1
Hence, if b1 is a root of b− n
m = 1 + b with 0 ≤ b1 ≤ 1. If b2 is a root of b− n
m = b − 1
with 1 ≤ b2. Then, the integration path is
b1 ≤ b ≤ b2
1 + b1 ≤ a ≤ b2 − 1
Then the limits are:
x1 : 1 + b1 ∼ 1 − b2
y1 : −b1 ∼ b2
ˆ x1 : −b1 ∼ b2
ˆ y1 : 1 + b1 ∼ 1 − b2
Then
m(P) =
1
2π2(2n(P3(ˆ x1) + P3(y1))|
b2
−b1 − 2m(P3(ˆ y1) + P3(x1))|
1−b2
1+b1)
=
2n
π2(P3(b2) − P3(−b1)) −
2m
π2 (P3(1 − b2) − P3(1 + b1))
If we write c1 = b
1
m
1 and c2 = b
1
m
2 , we have c1 is a root of
xn+m + xn − 1 = 0
with 0 ≤ c1 ≤ 1. Also c2 is a root of
xn+m − xn − 1 = 0
with 1 ≤ c2.
Then
m(P) =
2n
π2(P3(cm
2 ) − P3(−cm
1 )) −
2m
π2
µ
P3
µ
−
1
cn
2
¶
− P3
µ
1
cn
1
¶¶
=
2n
π2(P3(cm
2 ) − P3(−cm
1 )) +
2m
π2 (P3 (cn
1) − P3 (−cn
2))
77. What is next?
There are several open questions around this subject. We have been able to explain all
the known examples in three variables, except Condon’s example [3]
m(1 + x + (1 − x)(y + z)) =
28
5π2ζ(3) (16)
(John is working on it).
Also, we would like to generalize this construction to more variables. There are several
examples that we would like to explain, including
m((1 + x)(1 + y)(1 + w) + (1 − x)(1 − y)(1 − w)z) =
93
π4ζ(5) (17)
in [7].
The algebraic computation, although easier to perform than the integral, requires some
happy ideas that seem to be diﬀerent for each case. it would be nice to have an algorithm to
perform such decompositions. It would be nice being able to determine beforehand whether
it is possible to achieve such decompositions or not in each example. (These questions are
unknown for the two-variable case as well).
References
[1] D. W. Boyd, F. Rodriguez Villegas, Mahler’s measure and the dilogarithm (I), Canad.
J. Math. 54 (2002), 468 - 492.
[2] D. W. Boyd, F. Rodriguez Villegas, with an appendix by N. M. Dunﬁeld, Mahler’s
measure and the dilogarithm (II), (preprint, July 2003)
[3] J. Condon, Calculation of the Mahler measure of a three variable polynomial, (preprint,
October 2003)
[4] C. D’Andrea, M. Lal´ ın, On The Mahler measure of resultants in small dimensions. (in
preparation).
[5] A. B. Goncharov, Geometry of Conﬁgurations, Polylogarithms, and Motivic Cohomol-
ogy, Adv. Math. 114 (1995), no. 2, 197–318.
[6] A. B. Goncharov, The Classical Polylogarithms, Algebraic K-theory and ζF(n), The
Gelfand Mathematical Seminars 1990-1992, Birkhauser, Boston (1993), 113 - 135.
[7] M. N. Lal´ ın, Some examples of Mahler measures as multiple polylogarithms, J. Number
Theory 103 (2003), no. 1, 85–108.
[8] F. Rodriguez Villegas, Modular Mahler measures I, Topics in number theory (University
Park, PA 1997), 17-48, Math. Appl., 467, Kluwer Acad. Publ. Dordrecht, 1999.
[9] C. J. Smyth, On measures of polynomials in several variables, Bull. Austral. Math. Soc.
Ser. A 23 (1981), 49 - 63. Corrigendum (with G. Myerson): Bull. Austral. Math. Soc.
26 (1982), 317 - 319.
8[10] D. Zagier, Polylogarithms, Dedekind Zeta functions, and the Algebraic K-theory
of Fields, Arithmetic algebraic geometry (Texel, 1989), 391–430, Progr. Math., 89,
Birkhuser Boston, Boston, MA, 1991.
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An algebraic integration for Mahler measure

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An algebraic integration for Mahler measure

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