On the Mahler measure of resultants in small dimensions

Transcription

1 Journal of Pure and Applied Algebra On the Mahler measure of resultants in small dimensions Carlos D Andrea a, Matilde N. Lalín b, a Department d Álgebra i Geometría, Universitat de Barelona, Gran Via de les Corts Catalanes 585, 87 Barelona, Spain b Mathematial Sienes Researh Institute, 7 Gauss Way, Berkeley, CA 947, United States Reeived June 5; reeived in revised form 7 April 6 Available online 4 July 6 Communiated by A.S. Merkurjev Abstrat We prove that sparse resultants having Mahler measure equal to zero are those whose Newton polytope has dimension one. We then ompute the Mahler measure of resultants in dimension two, and examples in dimension three and four. Finally, we show that sparse resultants are tempered polynomials. This property suggests that their Mahler measure may lead to speial values of L-funtions and polylogarithms. 6 Elsevier B.V. All rights reserved. MSC: G55; G5; C8; 4Q99. Introdution Let A,..., A n Z n be finite sets of integral vetors, A i := {a i j },...,ki. We denote with Res A,...,A n Z[X,..., X n ] the assoiated mixed sparse resultant, whih is an irreduible polynomial in n + groups X i := {x i j ; j k i } of k i variables eah. It has the following geometri interpretation: onsider the system k i F i t,..., t n := x i j t a i j = i =,..., n of Laurent polynomials in the variables t,..., t n. Here t a stands for t a ta... ta n n where a = a,..., a n. The resultant Res A,...,A n vanishes on a partiular speialization of the x i j in an algebraially losed field K if the speialized system has a ommon solution in K \ {} n. See [3,3] for a preise definition of Res A,...,A n and some basi fats. Resultants are of fundamental importane for solving systems of polynomial equations and therefore have been extensively studied [3,4,7,9,5,3]. Reent researh has foused on arithmeti aspets of this polynomial suh as its height and its Mahler measure [5,,]. Reall that the absolute height of g := α α X α C[X,..., X n ] is defined as Hg := max{ α, α N k }, where k := k + + k n. Its logarithmi height is given by hg := log Hg = log max{ α, α N k }. Corresponding author. addresses: arlos@dandrea.name C. D Andrea, mlalin@math.ub.a M.N. Lalín. -449/$ - see front matter 6 Elsevier B.V. All rights reserved. doi:.6/j.jpaa.6.6.4

2 394 C. D Andrea, M.N. Lalín / Journal of Pure and Applied Algebra The Mahler measure of g is defined as mg := πi k log gx,..., X n dx T k X dx n X n, where for i =,..., n, dx i X i is an abbreviation for k dx i j x i j, and T k = {z,..., z k C k z = = z k = } is the k-torus. Some general relationships between the height and the Mahler measure are established in [8, Chapter 3] as well as [,]. In [] upper bounds for both the height and the Mahler measure of resultants are presented. However, very little seems to be known about the problem of expliitly omputing both the height and the Mahler measure of resultants. In the ase of heights, a first attempt was done in [5], where the heights of resultants in low degree and one variable are alulated. Jensen s formula gives a simple expression for the Mahler measure of a univariate polynomial as a funtion on its roots. However, it is in general a very hard problem to give an expliit losed formula for the Mahler measure of a multivariate polynomial. The simplest examples are Theorem.. [9, Example 5] m + x + y = 3 3 4π Lχ 3, = L χ 3,, where χ 3 h if h mod 3 Lχ 3, s := h h= s with χ 3 h := if h mod 3 if h mod 3 is the Dirihlet L-series in the odd harater of ondutor 3. Smyth also proved see [, Appendix ]: m + x + y + z = 7 π ζ3, where ζ denotes the Riemann zeta funtion. In this paper we fous on the expliit omputation of the Mahler measure of Res A,...,A n in the ase where the dimension of NRes A,...,A n the Newton polytope of Res A,...,A n is small. We assume that the family of supports A,..., A n is essential see [3, Se.], so Res A,...,A n is a polynomial of positive degree in the variables x i j. It is well-known see [8, Lemma 3.7] that we always have mres A,...,A n. The reason we fous on the dimension of the Newton polytope of the resultant and not on the number of variables and/or the size of the supports is due to some properties of the Mahler measure with respet to homogeneousness and hanges of variables. For instane, the Mahler measure of a homogeneous polynomial is the same as the Mahler measure of the orresponding dehomogenized polynomial. Moreover, Lemma. [, Lemma 7]. Let Py be a p-variable polynomial, and let V be a non-singular p p integer matrix, then mpy = mpy V, where y V denotes j yv j j,..., j yv pj j for y = y,..., y p and V = {v i j }. The whole situation may be summarized as follows: omputing the Mahler measure of a polynomial whose Newton polytope has dimension p is the same as omputing the Mahler measure of a p-variable polynomial. This is important beause we may expet different kinds of formulas aording to the number of variables meaning the dimension of the Newton polytope. For speulations onerning this matter, see []. 3

3 C. D Andrea, M.N. Lalín / Journal of Pure and Applied Algebra Evidene for this situation is Theorem in Setion whih states that Res A,...,A n has Mahler measure equal to zero if and only if its Newton polytope has dimension one, i.e. it is a segment. This result shows that Mahler measures and heights behave differently in resultants. For instane, if we set n =, A = {, }, A = {,,..., l}, then it turns out that Res A,A = ± l j x j x l j x j, j= and from here it is easy to see that hres A,A =. On the other hand, setting y j = j x j x l j x j, l mres A,A = m y j. j= The hange of variables is allowable, beause the x j are algebraially independent, so we may apply Lemma.. Dehomogeneizing, one obtains mres A,A = m + s + s + + s l, and this has been shown to be equal to logl + γ logl+ + O l+ as l, where γ is the Euler Masheroni onstant see [9], and also [8] for more estimates and generalizations. Moreover, it is still unknown how to haraterize all supports A,..., A n having hres A,...,A n =. In Setion we deal with sparse resultants having Mahler measure zero. Then we proeed to higher dimensions. In Setion 3 we fous on the ase where the Newton polytope of the resultant has dimension two or three. In Theorem, we ompute the Mahler measure of resultants in dimension two, and in Theorem 4, we show that omputing the Mahler measure of resultants in dimension three is essentially equivalent to the omputation of Mahler measures of univariate trinomials. In Theorem 6 we ompute the Mahler measure of trinomials having the same support. In Setion 4 we ompute the Mahler measure of a non trivial example in dimension four. All the omputations an be expressed in terms of linear ombinations of polylogarithms evaluated at algebrai numbers. From the point of view of Mahler measure, it is natural to wonder why we would expet resultants to be a soure of interesting examples of multivariate polynomials. In [6] Deninger established the relation between Mahler measure and regulators see also [7,]. More speifially, the Mahler measure of an irreduible polynomial P Q[s,..., s p ] is interpreted in terms of a speial value of the regulator ηs,..., s p in X, the projetive variety determined by {P = }. The regulator on the symbol {s,..., s p } K p MCX Q is initially defined in the ohomology of X \ {poles and zeros of s i}. A suffiient ondition for extending it to the ohomology of X is that the tame symbols of the faets are trivial. In that ase the polynomial is alled tempered [7]. If the symbol {s,..., s p } is trivial, then the tame symbols of the faets are trivial and ηs,..., s p is exat, and easily integrable by means of Stokes Theorem. This is the first step that may lead to a Mahler measure involving speial values of polylogarithms []. While the symbol is not neessarily trivial for a general polynomial, it is trivial for the ase of sparse resultants. This is the ontent of Setion 5. In Theorem 8, we show that resultants have trivial symbol, and so they are tempered polynomials. This fat suggests that the Mahler measure of resultants may be expressible in terms of ombinations of polylogarithms and that we might expet results in the style of the ones from Setions 3 and 4 to be held in more generality.. Resultants with Mahler measure equal to zero The main result of this setion is the following: Theorem. mres A,...,A n = dimnres A,...,A n =.

4 396 C. D Andrea, M.N. Lalín / Journal of Pure and Applied Algebra Proof. Assume first that dimnres A,...,A n =. We use Proposition 4. in [3], whih haraterizes all families of essential supports A,..., A n suh that the dimension of the Newton polytope is one: they must satisfy k i :=, i =,..., n. It turns out that [3, Proposition.] in this ase Res A,...,A n must be of the form ±X λ X λ, with λ, λ N k. It is very easy to see that polynomials of this kind have Mahler measure zero beause they are a monomial times the evaluation of T in another Laurent monomial. For the onverse, assume that mres A,...,A n =. Reall that the resultant is a primitive polynomial in Z[X,..., X n ]. By Kroneker s Lemma see for instane [8, Theorem 3.], Res A,...,A n must be a monomial times a produt of ylotomi polynomials evaluated in monomials. But Res A,...,A n is irreduible in C[X,..., X n ] as it is the equation of an irreduible surfae in the projetive omplex spae see [3, Lemma.]. Having its Mahler measure zero, the resultant must be of the form X α ± X β with α, β N k, i.e. a monomial times the polynomial T ± evaluated at another Laurent monomial. Hene, dimnres A,...,A n =. 3. The Mahler measure of resultants in dimensions two and three Now we would like to ompute the Mahler measure of the systems having dim NRes A,...,A n >. In order to do that, we first reall the following haraterization of the dimension of the Newton polytope of the resultant: Theorem 3. [3, Theorem 6.]. dimnres A,...,A n = k n, where, as defined in the introdution, k = n i= k i. We will ompute the Mahler measure of the resultants having dimnres A,...,A n =. By the previous theorem, this property only holds in the ase where there exists a unique i suh that k i = 3 and all other k i =, beause the k i must be greater than see [3, Theorem.]. Suppose w.l.o.g. that k = 3 and k = k = = k n =. Consider any linear transformation in SLn, Z whih maps the diretions in A i to multiples η i e i of the unit vetors for i =,..., n. After applying this transformation whih does not hange neither the Mahler measure nor the struture of Res A,...,A n, the original F i s defined in look as follows: F t,..., t n = x t a + x t a + x 3 t a 3, η F t,..., t n = x t x,... η F n t,..., t n = x n t n n x n. Let η := η + η + + η n. 4 Theorem. For systems having support as in 4, mres A,...,A n = η L χ 3,. Proof. It is straightforward to verify that the resultant of 4 is the following: for eah j =,..., n let ξ j run over the η j -roots of unity. Then, it turns out that Res A,...,A n equals, up to a monomial in the variables x, x,..., x n, n η x η xn ηn f, ξ,..., ξn ξ η j j = x ξ x Let V i := x i x i. By Lemma., M j := m f ξ η j j = x ξ x x η, ξ x x x n η,..., ξn xn. x n ηn

5 C. D Andrea, M.N. Lalín / Journal of Pure and Applied Algebra η = m f ξ V η, ξ V,..., ξ n V n ηn. ξ η j j = Now, sine mpt,..., t n = mpt p,..., t p n n by Lemma. one again, we have M j = m f ξ V, ξ V,..., ξ n V n. ξ η j j = Observe that oeffiients of absolute value one an be absorbed by variables, so M j = η j m f V, V,..., V n. Hene n M := mres A,...,A n = M j = ηm f V, V,..., V n. Now sine x, x, and x 3 are algebraially independent, we may replae x V a, x V a, and x 3 V a 3 by three independent variables W, W, and W, M = ηmw + W + W ; but this is just Smyth s result Theorem.: M = ηm + x + y = η 3 3 4π Lχ 3, = η L χ 3,. With the same proof as before, we an ompute the Mahler measure of more general systems as follows. Consider an essential system of the form F = x t a + x t a + + x l t a l, η F = x t x,... η F n = x n t n n x n. 5 Theorem 3. With the notation established above, for systems as 5 we have mres A,...,A n = η m + s + s + + s l. As mentioned in the introdution, the Mahler measure of polynomials of the form + s + s + + s p was estimated in [9] and later in [8]. Now we would like to ompute the Mahler measure of resultants having Newton polytope of dimension 3. Aording to Theorem 3., we must onsider essentially the following two senarios: k = 4, k = k = = k n =. This is a system of the form 5, and hene we have that mres A,...,A n = ηm + s + s + s 3 = η 7 π ζ3, by Smyth s result Theorem.. k = k = 3, k = k 3 = = k n =. This ase is treated below. Let k = k = 3 and k = k 3 = = k n =. Consider a linear transformation in SLn, Z whih maps the diretions in A i to multiples η i e i of the unit vetors for i =,..., n. F = x t a + x t a + x 3 t a 3, F = x t a + x t a + x 3 t a 3, η F = x t x,... η F n = x n t n n x n. 6

6 398 C. D Andrea, M.N. Lalín / Journal of Pure and Applied Algebra Let α i j Z be the first oordinate of the vetor a i j, i =,, j =,, 3, and set as before η := η +η 3 + +η n. Consider the following system of supports: A := {α, α, α 3 }, A := {α, α, α 3 }. 7 Observe that the ardinalities of A and A must be at least two, otherwise the family A, A,..., A n would not be essential. We get the following. Theorem 4. For systems like 6 we have mres A,...,A n = ηmres A,A. Proof. As in the proof of Theorem 6. in [3], it turns out that Res A,...,A n equals, up to a monomial fator, the produt of the Res A,A over all hoies of roots of unity. We an then follow the same lines as in the proof of Theorem and onlude the laim. Therefore the omputation of the Mahler measure of resultants in dimension three redues to the omputation of the Mahler measure of univariate systems like 7. Unfortunately, this does not seem to be very easy. In order to state our best result in that diretion, we need to reall some fats about polylogarithms see, for instane, [5]. Definition 5. The qth polylogarithm is the funtion defined by the power series Li q z := z j j q z C, z <. This funtion an be ontinued analytially to C\,. Observe that Li q = ζq and Li q = q ζq. In order to avoid disontinuities and to extend these funtions to the whole omplex plane, several modifiations have been proposed. We will only need the ases q =, 3. For q =, we onsider the Bloh Wigner dilogarithm: P z = Dz := ImLi z + log z arg z. 9 For q = 3, Zagier [5] proposes the following: P 3 z := Re Li 3 z log z Li z + 3 log z Li z. These funtions are one-valued, real analyti in P C \{,, }, and ontinuous in P C. Moreover, P q satisfies several funtional equations, the simplest ones being, for q =, D z = Dz, Dz = D z = D, z Dz = z D + D z z z z + D. z When z has absolute value one, Dz has a partiularly elegant expression: θ log sin t dt = De iθ = sin jθ j. 3 More about Dz an be found in [4]. For q = 3, we have, for instane, P 3 z = P 3 z, P 3 = P 3 z, 4 z P 3 z + P 3 z + P 3 = ζ3. 5 z 8

7 We are now ready to state our result: C. D Andrea, M.N. Lalín / Journal of Pure and Applied Algebra Theorem 6. Suppose that A = A, have both ardinality three. W.l.o.g. we an suppose that A = {, p, q}, with p < q and gdp, q =. Then, mres A,A = pp3 π ϕ q q P 3 ϕ p + pp 3 φ q + q P 3 φ p where ϕ is the real root of x q + x q p = suh that ϕ, and φ is the real root of x q x q p = suh that φ. Proof. All along this proof, we will write Res as short of Res {,p,q},{,p,q}. First, we will show that ResA + Bt p + t q, C + Et p + t q = C A q E A BC p B E q p. Let us set f := A + Bt p + t q and g := C + Et p + t q. By using [3, Ex 7 Chapter 3] we see that Res f, g = Res f, g f = ResA + Bt p + t q, C A + E Bt p. 6 Let ξ be a primitive pth root of the unity, then all the roots of C A + E Bt p are ξ j C A B E p, j =,..., p. By using the Poisson produt formula for the omputation of Res see display.4 in Chapter 3 of [3], we onlude that 6 equals qp E B q p f ξ j C A B E The last produt in 7 is of the form p α βξ j = α p β p p p = qp E B q A + B C A B E + ξ q j C A B E with α = BC AE B E and β = C A B E q p this is due to the fat that gdp, q =. So we get that 7 equals BC AE qp E B q p p C Aq B E p B E q = qp q BC AE p B E q p p+q C A q = qp+p+q+ C A q AE BC p B E q p. The laim holds straightforwardly by noting that qp + q + p + = q + p + is even if gdp, q =. Now we have to ompute the Mahler measure of C A q E A BC p B E q p. q p. After setting C = C A, E = E B and dividing by A p, we see that it is enough to onsider the polynomial A q p C q B q E C p E q p. Now set Z = A q p B q and divide by B q. We need to ompute mzc q E C p E q p. 8 By using Jensen s equality with respet to the variable Z and the fat that mc q =, we dedue that 8 equals πi log + E q p E C p dc de T C q, C E where log + x = log x for x and zero otherwise. 7

8 4 C. D Andrea, M.N. Lalín / Journal of Pure and Applied Algebra Now write C = E Y. The expression above simplifies as follows: πi log + E q p Y p dy de T Y E q. Y E Setting Y = e iα, E = e iβ, we have that this expression an be omputed as follows: π log + sin p α sin q p β π π sin q α + β dα dβ = π log + sin p α sin q p β π sin q α + β dα dβ. 9 For π β, set γ = β. We an then simplify 9: = π log + sin p α sin q p β sin q α + β dα dβ + π Now we perform a hange of variables. For the first term, write a = sin α sinα + β, b = sin β sinα + β, then dαdβ = da db a b. For the seond term, set then a = sin α sinα γ, b = sin γ sinα γ, dαdγ = da a db b. log + sin p α sin q p γ sin q α γ dα dγ. This hange of variables has a geometri interpretation: we an think of a and b as the sides of a triangle whose third side has length one. The side of length a is opposite to the angle α and the side of length b is opposite to β. This onstrution is possible beause of the Sine Theorem. Fig. desribes how the sides vary aording to the angles. The integral beomes the sum of four terms, eah of them orresponding to eah ase in Fig.. π +b log + a p b q p da b a + b log + a p b q p da a = π π b +b b log + a p b q p da a db b + π db b + π db b + +b b π +b b log + a p b q p da a +b db b log + a p b q p da +b a log + a p b q p da a Now write q p = a and d p = b. Then the previous expression redues to p q p π +d p q p d p q p = p q p π I + I. log + d d dd d + p q p π db b. +d p q p d p q p db b log + d d Let us ompute I. Sine the argument has the term log + d, we need to restrit the domain to the ase d. Now observe that sine d, then d q p d p. On the other hand, + d p d q p if and only if d q + d q p. dd d

9 C. D Andrea, M.N. Lalín / Journal of Pure and Applied Algebra Fig.. a Case when b in the first integral. b Case when b in the first integral. Case when γ α in the seond integral. d Case when γ α in the seond integral. For future referene, let ϕ be the unique root of x q + x q p = in [, ]. I = = = +d p q p ϕ ϕ ϕ d logd d log d + d p q p dd d dd d = ϕ log d log d + log d log + d p + log + d p d q pd q p d The first term in the integral is easy to integrate: ϕ log d d dd = log3 ϕ. 3 l= +d p q p For the seond term, we use the series expansion of log + x: log d log + d p dd = l d pl log ddd ϕ d ϕ l l= = l d pl pl log d + l d pl ϕ pl dd ϕ d dd d dd. l= = log ϕ p Li ϕ p + p Li 3 Li 3 ϕ p. We apply definition to onlude that this expression equals p P 3 ϕ p + q p log3 ϕ 3 Finally, we ompute the third term of ϕ log + d p dd = d p ϕ q p 3 4p ζ3. log d

10 4 C. D Andrea, M.N. Lalín / Journal of Pure and Applied Algebra setting = Then = p +d p ϕ q p log = q p3 log 3 ϕ 3p + d = q + pq p log 3 ϕ 3p = q + pq p log 3 ϕ 3p log Li p I = + log3 3p p log log log3 3p log3 3p p + p Li 3ϕ q p p Li 3 = p P 3ϕ q p q p log 3 ϕ 3 pq p P 3ϕ q p p P 3 l= ϕ q p + p l l log ϕ q p ϕ q p log log + p q p log ϕli ϕ q p p. p q p P 3 ϕ p pq p P 3 ϕ q p l= l l d 3ζ3 4p q p. Let us ompute I. As before, we need to restrit our domain to the ase d. Sine d, we have d q p + d p always. On the other hand, d p d q p if and only if d q d q p. For future referene, let φ be the unique root of x q x q p = in [,. Then I = φ +d p q p d We proeed to ompute I, I = φ setting = d = q p φ logd d log d + d p q p dd d + +d p q p φ d p q p logd d dd d = I + I. dd d = log + p q log d q p φ q log q log log + p + log + p d. Using similar omputations to those from I, we obtain + I = q log 3 φ q log φ log 6q p + 3q p + φ p + pq p P 3 + For the ase of I we have: I = φ log φ log + pq p P 3 log d + d p q p log dd p 6q p + q p + 3qζ3 4p q p. dd d d q p q p P 3

11 = C. D Andrea, M.N. Lalín / Journal of Pure and Applied Algebra φ log + p q p log p + q log log p q log log + p d setting = d. Now we ompute eah of the terms in Eq. : φ φ log + p d = log p d = p log + log φ 3 φ q log f f d f p P 3 φ p + setting f = p = p log f log f + log f log f d f φ q p φ q f = q log 3 φ q p log φli + p ζ3 p Li 3 = q log 3 φ 3 p P 3 φ φ φ q φ q + p ζ3. log log p d = log φ p Li = p P 3 log log + p d = log φ p Li = p P 3 Putting all the terms together, log q p +φ I = p log φ 6 φ p and hene, p P 3 q p I = p P 3 + Now we an onlude: I + I = pq p P 3ϕ q p + pq p P 3 + p P 3 φ q + p Li 3 q log3 φ. 3 + p Li 3 log φ log + 3 q log 3 φ 6 φ q + + p ζ3 q log φ log + + q p P qζ3 4p + p P 3 φ q + q p P 3. p q p P 3 ϕ p 3ζ3 4pq p + pq p P 3 Let us note that P 3 + P 3 = ζ3 beause of Eq. 5, hene, P 3 = 7 8 ζ3.. q p P 3 φ q + q p q p P 3 φ p.

12 44 C. D Andrea, M.N. Lalín / Journal of Pure and Applied Algebra so Then I + I = pq p P 3ϕ q p ζ3 pq p + pq p P 3 p q p P 3 ϕ p φ q + q p q p P 3 φ p. Now, we use Eq. 5 again in order to obtain P 3 ϕ q + P 3 ϕ q + P 3 ϕ q = ζ3 from where P 3 ϕ q + P 3 ϕ q p + P 3 ϕ p = ζ3, P 3 ϕ q p = ζ3 P 3 ϕ p P 3 ϕ q. Hene, we have I + I = pq p P 3ϕ q whih proves our laim. 4. An example in dimension 4 q p q p P 3 ϕ p + pq p P 3 φ q + q p q p P 3 φ p, We would like to study one more example, whih is a partiular ase of a 4-dimensional resultant. Let us set n = and A = A = A = A := {,,,,, }. We will use a formula due to Cassaigne and Maillot. Theorem 4. [3, Proposition 7.3.]. { a D e iγ + α log a + β log b + γ log πma + bx + y = b π log max{ a, b, } if if not where stands for the statement that a, b, and are the lengths of the sides of a triangle; and α, β, and γ are the angles that are opposite to the sides of lengths a, b and respetively. Theorem 7. mres A,A,A = 9ζ3 π. Proof. In order to simplify the notation, we will use the variables a, b,,... instead of the x i j s. a b Res A,A,A = det d e f. g h i Now, let us proeed to eliminate homogeneous variables: a b m d e f = m d e f = m e f g h i g h i h i = m e f = me i f h. h i

13 C. D Andrea, M.N. Lalín / Journal of Pure and Applied Algebra Fig.. We always obtain a triangle for this ase. Let us observe that mx y z w = mx y + zw + z x. 3 Hene we an think of the polynomial x y + zw + z x as a linear polynomial in the variables y, w, whose oeffiients are in Z[x, z]. Beause of the iterative nature of the definition of Mahler measure, we an hoose to integrate first with respet to the variables y and w, regarding x and z as parameters. If we do that, we obtain formula with the sides of the triangle equal to x, z, and z x. Now in order to ompute the Mahler measure, we still need to integrate this formula with respet to x and z. Set x = e iα and z = e iβ. This notation is onsistent with the names for the angles of the triangle beause of the Sine Theorem see Fig.. We obtain that 3 is equal to π 3 β D sin β sin α eiβ α First, we integrate the terms involving logarithms: α α log sin α dβdα = = + α log sin α + β α log sinβ α + π β log sin β dα dβ. απ α log sin α dα Beause of πα α os jα dα = πα α sin jα π j j os jα π = π α 4 j 3 we onlude that Then απ α log sin α dα = πζ3. β π β log sin β dαdβ = απ α log e iα dα = απ αre by analogy with the ase of α. By setting γ = β α in the third logarithmi term, β β α log sinβ α dαdβ = = sin jα π α j dα os jα 4 j 3 dα = π j 3, βπ β log sin β dβ = πζ3, γ γ log sin γ dαdγ γ π γ log sin γ dγ = πζ3. e jiα j dα.

14 46 C. D Andrea, M.N. Lalín / Journal of Pure and Applied Algebra On the other hand, we need to evaluate β sin β D sin α eiβ α dαdβ. Using Eq., sin β z D sin α eiβ α = D = x = z D + Dx + D x De iβ α + De iα + De iβ. z 4 Then integral 4 is the sum of three terms. We proeed to ompute eah of them: α De iα dβdα = = π αde iα dα π α sin jα j dα. But π α Then sin jα j dα = π α π αde iα dα = πζ3. os jα j 3 The other terms an be omputed in a similar fashion: β β De iβ dα dβ = De iβ α dα dβ = Thus, we onlude that γ π mx y z w = π 3 = β De iβ dβ = πζ3, π De iγ dαdγ os jα j 3 dα = π j 3. π γ De iγ dγ = πζ3. 3 πζ3 + 3 πζ3 = 9ζ3 π. 5. Resultants are tempered polynomials In this setion we will leave the elementary approah given above and turn instead to study algebrai properties of resultants and Mahler measures in the ontext of Milnor K -theory. We will sketh the relation here and we refer to [6], [7], and [] for preise details. For P C[s,..., s p ] irreduible, we write Ps,..., s p = a i s,..., s p s i p. i Let P s,..., s p = a i s,..., s p, the main non-zero oeffiient with respet to s p. Let X be the zero set {Ps,..., s p = }.

15 C. D Andrea, M.N. Lalín / Journal of Pure and Applied Algebra By applying Jensen s formula to the Mahler measure of P with respet to the variable s p, it is possible to write mp = mp iπ p log s p ds ds p Γ s s p where Γ = {Ps,..., s p = } { s = = s p =, s p }. From this formula, Deninger [6] establishes mp = mp iπ p ηs,..., s p 5 Γ where ηs,..., s p is a ertain Rp -valued smooth p -form in X C \ S. Here, S denotes the set of poles and zeros of the funtions s,..., s p. For example, in two variables, η has the following shape: ηx, y = log y i d arg x log x i d arg y. η is a losed form that is multipliative and antisymmetri in the variables s,..., s p. Therefore, it is natural to think of η as a funtion on p CX Q we tensorize by Q beause η is trivial in torsion elements. Even more, η s, s, s 3,..., s p is an exat form. We may also see ηs,..., s p as a lass in the DeRham ohomology of X \ S. This situation allows us to think of the ohomologial lass of η as a funtion in the Milnor K -theory group K p M CX Q. Reall that for a field F the Milnor K -theory group is given by K M p F := p F / s s s p, s i F. If we an extend this lass to the ohomology of X, η beomes a regulator. In ertain ases, seeing the Mahler measure as a regulator allows us to explain its relation to speial values of L-funtions via Beilinson s onjetures and similar results. The ondition that the lass of ηs,..., s p be extended to X is given by the triviality of the tame symbols in the Milnor K -theory. A stronger ondition is that ηs,..., s p is exat. Sine η is defined in K p M CX Q, ηs,..., s p is exat if the symbol {s,..., s p } is trivial in K p M CX Q. This is a very speial ondition that is not true for a general polynomial. However, it is true for resultants: Theorem 8. The symbol is trivial. {x,..., x k,..., x n,..., x nkn } K M k CX Q 6 In order to prove Theorem 8, we will need the following Lemma 9. Consider a p-variable polynomial Pu,..., u p := x i u i. Here i is a multiindex, u i = u i u i p p. Let E be a field ontaining the x i s and let α,... α p be in E. Then x i is of the form Pα,..., α p a i + i,h r i,h α i i,h + j,l s j,l b j b j b j,l, where a i, b j, b j,l, i,h are elements of E and r i,h, s j,l are integers. Proof. First we prove the ase for whih p =. Let us write Pα = x + x α + x α... + x q α. x x x q

16 48 C. D Andrea, M.N. Lalín / Journal of Pure and Applied Algebra and After setting y = x and y i = x i x i for i >, we obtain Pα = y + y α + y α... + y q α, x x q = y y q. We may introdue α in the last plae of the wedge produt of y i : y y q = y y q y q α y y q α. Now we introdue α in the seond to the last plae = y y q + y q α y q α y + y q α y q α y y q α = y y q y q α + y q α y q α y y q α y q y + y q α y q α y y q α. Then we introdue α in the third to the last plae. We ontinue in this fashion = y y q + y q α + y q α y q α + y q α y q α y + y q α + y q α y q α + y q α y q α y y q α y q y + y q α y q α y y q α. After q more steps, we get Pα q a i + ± b h b h b h,l h= q y y h α y h+ y q, whih proves the laim in this ase. Let us now onsider p >. We use indution on p. Suppose that the laim is true for p. Then we may write Pα,..., α p = P i α,..., α p α i p. h= By the indutive hypothesis we obtain xi = P α,..., α p P h α,..., α p a i + i,h r i,h α i i,h + j,l s j,l b j b j b j,l. Now apply the ase p = to the first term in order to obtain the desired result. In the notation of this proof we used that the oeffiients were all different from zero. The proof is easily adapted to the ase when some oeffiients are zero. We are going to use this lemma in full generality. Proof Theorem 8. By definition of Milnor s K -theory, it is suffiient to work in k CX Q. Consider the variety Y = {z Res A,...,A n = } C k+. Let H := CY and E be a finite extension of H ontaining all the roots that are ommon to the last n polynomials F,..., F n in an algebraially losed field ontaining H. We will prove that the symbol determined by z x j x j x nj is trivial in k+ E Q. This fat implies that the orresponding symbol in Kk+ M E Q is trivial. Now j : H E indues j : K MH K ME whose kernel is finite see []. Then K MH Q K M E Q is injetive and we onlude that the symbol must be trivial in k+ H Q.

17 C. D Andrea, M.N. Lalín / Journal of Pure and Applied Algebra The triviality of this symbol implies that x j x j x nj is trivial in s CX Q, beause it is the image by the tame symbol morphism with respet to the valuation determined by z =, [4] in the language of Newton polytopes, Res A,...,A n orresponds to a faet of z Res A,...,A n. We will use the Poisson produt formula for sparse resultants Theorem. in [6] and its refinement in [5, Theorem 8]: Res A,...,A n = d Res η η F α, 7 η α where η runs over all the maximal faets of the Newton polytope assoiated to the Minkowski sum A + + A n, Res η stands for the faet resultant assoiated to and the faet η, d η is a non negative integer, and α runs over all the ommon solutions of the system F = = F n = in E \ {} n. Let us proeed by indution on n. For n = we have, F = x t a + + x k t a k and F = x t a + + x k t a k, where we assume w.l.o.g. that a < a < < a k and a < a < < a k. Also, we an suppose w.l.o.g. that a = a =. This is due to the fat that the resultant is invariant under translations of the A i s. Then, ResF, F = x d k F α, where α runs over the roots of F in E \ {}, and d is a positive integer. Let F := x t a + + x k t a k. Then z x j x j = F α k x j F α k j= x j + dx k k x j F α The seond term is zero beause it ontains two opies of x k. By applying Lemma 9, the first term yields a ombination of terms of the form b j b j b j,l, whih is trivial in K -theory. Now let n >, and assume again w.l.o.g. that all the supports A i are ontained in N n, so the F i s are Taylor polynomials and we an use Lemma 9. Fix a solution α = α..., α n in E \ {} n for the last n equations. As in the ase n =, it is easy to see that we may write equations of the kind x l = a F l α where a is equal to a produt possibly empty of α i. We need to onsider z x j x j x nj, whih, due to 7, equals F α + η k d η Res η x j F α k k j= x j F α x j F n α k j= k n x nj j= x j F n α The sum is over all the solutions α of F i = for i n. For the first term, we apply Lemma 9 to eah of the n + sets of oeffiients and obtain a ombination of terms of the form b j b j b j,l, whih is trivial in K -theory. The terms that orrespond to ombinations of α i i,h are zero beause we have n different α i s but n + terms, whih means that some α i appears twie in the wedge produt and then it must be equal to zero. Consider the seond term. For eah η and i =,..., n, we define G η i := F η i, the restrition of F i to the faet η see [6] for a preise definition of these polynomials. As the oeffiients of the G η i s are inluded in the oeffiients of the F i, we an apply indution and obtain the triviality of this term. This an be done due to the fat that Res η is always a resultant assoiated to a system of dimension n or less. Hene, the symbol is trivial in Kk+ M E Q and that proves the laim. k n j= x nj. k j= x j.

18 4 C. D Andrea, M.N. Lalín / Journal of Pure and Applied Algebra Aknowledgements The authors wish to thank Fernando Rodriguez-Villegas for several suggestions and also Martín Sombra for explaining to them his work []. They would also like to express their gratitude to the Miller Institute at UC Berkeley and the Department of Mathematis at the University of Texas at Austin for their support. Matilde Lalín is also grateful to the Harrington fellowship and John Tate for his or her support. The authors are grateful to the Referee for their suggestions. Referenes [] H. Bass, J. Tate, The Milnor ring of a global field, in: Algebrai K -Theory, II: Classial Algebrai K -Theory and Connetions with Arithmeti Pro. Conf., Seattle, Wash., Battelle Memorial Inst., 97, in: Leture Notes in Math., vol. 34, Springer, Berlin, 973, pp [] D. Boyd, Speulations onerning the range of Mahler s measure, Canad. Math. Bull [3] D. Cox, J. Little, D. O Shea, Using Algebrai Geometry, in: Graduate Texts in Mathematis, vol. 85, Springer-Verlag, New York, 998. [4] C. D Andrea, Maaulay style formulas for sparse resultants, Trans. Amer. Math. So [5] C. D Andrea, K. Hare, On the height of the Sylvester resultant, Experiment. Math [6] C. Deninger, Deligne periods of mixed motives, K -theory and the entropy of ertain Z n -ations, J. Amer. Math. So [7] I.Z. Emiris, B. Mourrain, Matries in elimination theory, in: Polynomial elimination algorithms and appliations, J. Symboli Comput [8] G. Everest, T. Ward, Heights of polynomials and entropy in algebrai dynamis, in: Universitext, Springer-Verlag London, Ltd, London, 999. [9] A. Khetan, The resultant of an unmixed bivariate system, in: International Symposium on Symboli and Algebrai Computation, ISSAC Lille, J. Symboli Comput [] T. Krik, L.M. Pardo, M. Sombra, Sharp estimates for the arithmeti Nullstellensatz, Duke Math. J [] M. Lalín, Some examples of Mahler measures as multiple polylogarithms, J. Number Theory [] M. Lalín, An algebrai integration for Mahler measure, preprint. August 5. [3] V. Maillot, Géométrie d Arakelov des variétés toriques et fibrés en droites intégrables, Mém. So. Math. Fr. N.S [4] J. Milnor, Algebrai K -theory and quadrati forms, Invent. Math / [5] M. Minimair, Sparse resultant under vanishing oeffiients, J. Algebrai Combin [6] P. Pedersen, B. Sturmfels, Produt formulas for resultants and Chow forms, Math. Z [7] F. Rodriguez-Villegas, Modular Mahler measures I, in: Topis in Number Theory University Park, PA 997, in: Math. Appl., vol. 467, Kluwer Aad. Publ., Dordreht, 999, pp [8] F. Rodriguez-Villegas, R. Toledano, J. Vaaler, Estimates for Mahler s measure of a linear form, Pro. Edinb. Math. So [9] C.J. Smyth, On measures of polynomials in several variables, Bull. Austral. Math. So Corrigendum: G. Myerson; C.J. Smyth, [] C.J. Smyth, An expliit formula for the Mahler measure of a family of 3-variable polynomials, J. Th. Nombres Bordeaux [] M. Sombra, Estimaiones para el Teorema de Ceros de Hilbert, Tesis de Dotorado, Departamento de Matemátia, Universidad de Buenos Aires, 998. [] M. Sombra, The height of the Mixed Sparse resultant, Amer. J. Math [3] B. Sturmfels, On the Newton polytope of the resultant, J. Algebrai Combin [4] D. Zagier, The Dilogarithm funtion in Geometry and Number Theory, in: Number Theory and Related Topis, Tata Inst. Fund. Res. Stud. Math [5] D. Zagier, Polylogarithms, Dedekind zeta funtions and the algebrai K -theory of fields, in: Arithmeti Algebrai Geometry Texel, 989, in: Progr. Math., vol. 89, Birkhauser Boston, Boston, MA, 99, pp