Closed sets of Mahler measures

Abstract

Abstract. Given a k-variable Laurent polynomial F , any ℓ×k integer matrix A naturally defines an ℓ-variable Laurent polynomial F A . I prove that for fixed F the set M(F ) of all the logarithmic Mahler measures m(F A ) of F A for all A is a closed subset of the real line. Moreover, the matrices A can be assumed to be of a special form, which I call Saturated Hermite Normal Form. Furthermore, if F has integer coefficients and M(F ) contains 0, then 0 is an isolated point of this set.I also show that, for a given bound B > 0, the set M B of all Mahler measures of integer polynomials in any number of variables and having length (sum of the moduli of its coefficients) at most B is closed. Again, 0 is an isolated point of M B .These results constitute evidence consistent with a conjecture of Boyd from 1980 to the effect that the union L of all sets M B for B > 0 is closed, with 0 an isolated point of L.

Mahler measures of polynomials that are sums of a bounded number of monomials

“…He conjectured that L is a closed subset of R. Our Theorem 2 below is a result in the direction of this conjecture, but where we restrict the polynomials F under consideration to be the sum of at most k monomials. In [13,Theorem 3], the second author proved another restricted closure result of this kind, where the restriction was, instead, to integer polynomials F of bounded length (sum of the moduli of its coefficients).…”

“…Proof. The first part follows from results of Boyd [2, p. 118] and Lawton [6]; see also [13,Lemma 13 and Proposition 14]. Next, note that F is the sum of k monomials of the form c(j)z j ℓ , so that F r is the sum of k monomials of the form c(j)(z r ) j = c(j)z rj = a i z t i say, for some i, where j ∈ J is a column vector.…”

“…Finally, to show that 0 is an isolated point of m(H k ), note that, by [13,Theorem 2] it is an isolated point of every M(a 1 z 1 + · · · + a k z k ) that contains 0. Hence, since 0 ∈ U B for every B > 0, it is an isolated point of U B and therefore also of m(H k ).…”

Mahler's measure and elliptic curves with potential complex multiplication

“…Moreover, Mahler's measure of polynomials in one variable is linked to the logarithmic Weil height of algebraic numbers (see [6,Proposition 1.6.6]) and therefore to questions of Diophantine nature, such as Lehmer's problem. This asks whether or not the set {m(P) : P ∈ Z[x]} \ {0} ⊆ R >0 has a minimum, and it seems that it could be approached by studying the Mahler measure of polynomials in multiple variables (see [68]). These Mahler measures appear to be far more mysterious than their one-variable counterparts: in particular, they appear to be related to special values of L-functions.…”

Limits of Mahler measures in multiple variables

“…Moreover, Smyth [60] used Lawton's result to show that the set M can be written as a nested ascending union of closed subsets of R. In fact, Smyth proves more generally that for every Laurent polynomial P ∈ C[z ±1 1 , . .…”