Examples of Mahler Measures as Multiple Polylogarithms

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1 Examples of Mahler Measures as Multiple Polylogarithms The many aspects of Mahler s measure Banff International Research Station for Mathematical Innovation and Discovery BIRS, Banff, Alberta, Canada April 26 May, 23 Matilde N. Lalín. Mahler Measure Definition For P C[x ±,...,x± n ], the logarithmic Mahler measure is defined by mp := =... 2πi n log Pe 2πiθ,...,e 2πiθn dθ...dθ n T n log Px,...,x n dx x... dx n x n 2 Jensen s formula provides a simple expression for the Mahler measure in the one-variable case. The several-variable case is more complicated. Many examples with explicit formulae have been produced. See [4], [], [2], [3], [4], [5], [6] The simplest example with two variables is due to Smyth [3]: Where m + x + y = 3 3 4π Lχ 3, 2 = L χ 3, 3 Lχ 3, s := n= χ 3 n n s is the L-series in the character of conductor 3: if n mod 3 χ 3 n = if n mod3 otherwise The analogous example with three variables is also due to Smyth: m + x + y + z = 7 2π2ζ3 4 The general linear case with two variables is due to Cassaigne and Maillot, [] : for a, b, c C, D a b e iγ + α log a + β log b + γ log c πma + bx + cy = 5 π log max{ a, b, c } not Here stands for the statement that a, b, and c are the lengths of the sides of a triangle, and α, β, and γ are the angles opposite to the sides of lengths a, b, and c respectively. See picture.

2 a γ β b c α D stands for the Bloch Wigner dilogarithm see definition later. The term with the dilogarithm can be interpreted as the volume of the ideal hyperbolic tetrahedron which has the triangle as basis and the fourth vertex is infinity see [], [7]. 2. Examples of higher weight We have obtained [9] examples of polynomials in several variables whose Mahler measures depend on polylogarithms. The first column of the table shows the polynomials. Here α is a complex number different from zero. The second column indicates the values of the first column for the case α =. πm + y + α yx 2Lχ 4, 2 π 2 m + w + y + α w yx π 3 m + v + w + y + α v w yx 7ζ3 7πζ3 + 4 j<k j 2j+ 2 k 2 π 2 m + x + αy + z 7 2 ζ3 π 3 m + w + x + α wy + z π 4 m + v + w + x + α v wy + z 2π 2 Lχ 4, j<k 93ζ5 j+k+ 2j+ 3 k π 2 7 π2 m + w + y + wx y 2ζ3 + 2 log 2 Here χ 4 is the real odd character of conductor 4, i.e. if n mod 4 χ 4 n = if n mod4 otherwise Let us observe that all the presented formulae share a common feature. If we assign weight to any Mahler measure and to π, then all the formulae are homogeneous, meaning all the monomials have the same weight, and this weight is equal to the number of variables of the corresponding polynomial. 3. Polylogarithms 2

3 We need the following definitions see [6], [7], [8] Definition 2 Multiple polylogarithms are defined as the power series Li k,...,k m x,...,x m := x <n <n 2 <...<n m n k nk 2 n xn xnm m 2...nkm m 6 which are convergent for x i <. The weight of a polylogarithm function is the number w = k k m and its length is the number m. Definition 3 Hyperlogarithms are defined as the iterated integrals am+ I k,...,k m a :... : a m : a m+ := dt dt t a t... dt dt dt t t a }{{} 2 t... dt dt... dt t t a }{{} m t... dt t }{{} k k 2 k m where k i are integers, a i are complex numbers, and bl+ dt dt dt... =... t b t b l t... t l b l+ t b dt l t l b l The value of the integral above only depends on the homotopy class of the path connecting and a m+ on C \ {a,...,a m }. It is easy to see that I k,...,k m a :... : a m : a m+ = m a2 Li k,...,k m, a 3 a m,...,, a m+ a a 2 a m Li k,...,k m x,...,x m = m I k,...,k m :... : x... x m a m x m : which gives an analytic continuation to multiple polylogarithms. For instance, with the convention about integrating over a real segment, simple polylogarithms have an analytic continuation to C \ [,. There are modified versions of these functions which are analytic in larger sets, like the Bloch-Wigner dilogarithm, Dz := ImLi 2 z + log z arg z z C \ [, 7 which can be extended as a real analytic function in C \ {, } and continuous in C. 4. General method for building examples The idea behind the computations that led to the examples in the table, is the following.. Let P α C[x,...,x n ] whose coefficients depend polynomially on a parameter α C. For instance, start with P α x = + αx, whose Mahler measure is log + α. 2. We replace α by α y +y and obtain a polynomial P α C[x,...,x n, y]. In the example, P α x, y = + y + α yx. 3

4 3. The Mahler measure of the second polynomial is a certain integral of the Mahler measure of the first polynomial. m P α = dy m P 2πi α y in the example, = log + T +y y 2πi α y dy T + y y 4. If the Mahler measure depends just on the absolute value of α, we can make a change of variables u = α y +y to be precise, first write y = e iθ and then set u = α tan θ 2. We obtain, mp u mp u du m P α = 2 π In the example, α du u 2 + α 2 = i π m + y + α yx = i π = i π = i π I 2 i i α : I 2 α : log + u s dt t u + i α u i α u + i α s + i α u i α s i α du ds = i π Li 2i α Li 2 i α If we look back at the table, we have splitted the examples into three families. The first family was developed starting from +αx, the second family starts from +x+αy +z see [2], [6]. The last polynomial was obtained by integrating one particular case of Maillot s formula: + αx + αy. 5. The example with five variables By applying the method described above, we can prove that π 4 m + v + w + x + v wy + z = 7π 2 ζ3 + 8Li 3,2, Li 3,2, +8Li 3,2, Li 3,2, 8 The values Li 3,2 ±, ± are alternating Euler sums. We use formula 75 of [], which in this particular case, states that Li 3,2 x, y = 2 Li 5x y + Li 3 xli 2 y + 3Li 5 x + 2Li 5 y Li 2 x yli 3 x + 2Li 3 y for x, y = ±. Taking into account that we get Li k = ζk and Li k = 2 k ζk 9 Li 3,2, Li 3,2, + Li 3,2, Li 3,2, = 2 4 ζ2ζ ζ5 We obtain the result by using that ζ2 = π2 6 4

5 References [] J. M. Borwein, D. M. Bradley, D. J. Broadhurst, Evaluations of k-fold Euler/Zagier sums: a compendium of results for arbitrary k, Electronic J. Combin , no. 2, #R5. [2] D. W. Boyd, Speculations concerning the range of Mahler s measure, Canad. Math. Bull , [3] D. W. Boyd, Mahler s measure and special values of L-functions, Experiment. Math , [4] D. W. Boyd, F. Rodriguez Villegas, Mahler s measure and the dilogarithm I, Canad. J. Math , [5] G. Everest, T. Ward, Heights of Polynomials and Entropy in Algebraic Dynamics, Universitex UTX, Springer-Verlag London 999. [6] A. B. Goncharov, The Classical Polylogarithms, Algebraic K-theory and ζ F n, The Gelfand Mathematical Seminars , Birkhauser, Boston 993, [7] A. B. Goncharov, Polylogarithms in arithmetic and geometry, Proc. ICM-94 Zurich 995, [8] A. B. Goncharov, Multiple polylogarithms and mixed Tate motives Preprint. [9] M. N. Lalín, Some examples of Mahler measures as multiple polylogarithms Preprint. [] V. Maillot, Géométrie d Arakelov des variétés toriques et fibrés en droites intégrables. Mém. Soc. Math. Fr. N.S. 8 2, 29pp. [] J. Milnor, Hyperbolic Geometry, The First 5 Years.Bulletin, AMS 982 vol. 6, nbr. pp [2] F. Rodriguez Villegas, Modular Mahler measures I, Topics in number theory University Park, PA Acad. Publ. Dordrecht, 999. [3] C. J. Smyth, On measures of polynomials in several variables, Bull. Austral. Math. Soc. Ser. A 23 98, Corrigendum with G. Myerson: Bull. Austral. Math. Soc , [4] C. J. Smyth, An explicit formula for the Mahler measure of a family of 3-variable polynomials, J. Th. Nombres Bordeaux 4 22, [5] C. J. Smyth, Explicit Formulas for the Mahler Measures of Families of Multivariavle Polynomials. Preprint [6] S. Vandervelde, A formula for the Mahler measure of axy + bx + cy + d Preprint. [7] Notes of the course Topics in K-theory and L-functions by F. Rodriguez Villegas. villegas/course.html 5