On the Mahler measure of resultants in small dimensions

Abstract

We prove that sparse resultants having Mahler measure equal to zero are those whose Newton polytope has dimension one. We then compute 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 special values of L-functions and polylogarithms.Comment: 2 figure

Mahler measure of $P_d$ polynomials

“…This interesting question is suggested by Brunault. D'Andrea and Lalín in [8], defined a polynomial in 4 variables which we call it P ∞ , and P ∞ := (x − 1)(y − 1) − (z − 1)(w − 1), they proved m(P ∞ ) = 9 2π 2 ζ(3). Therefore lim d→∞ m(P d ) = m(P ∞ ), which remind us of the theorem of Boyd-Lawton.…”

Limits of Mahler measures in multiple variables

“…Finally, D'Andrea and Lalín [18,Theorem 7] have proved that m(P ∞ ) = −18 • ζ (−2), which yields back the convergence (38) proved by Mehrabdollahei in [47].…”

An algebraic integration for Mahler measure

“…We will proceed to the study of a family of three-variable polynomials that come from the world of resultants, namely, Res {0,m,m+n} . This family was computed in [DL06] and the computation is quite involved, though elementary. The Mahler measure of Res {0,m,m+n} is the same as the Mahler measure of a certain rational function.…”

“…Here is an example. We will study the case of Res {(0,0),(1,0),(0,1)} , whose Mahler measure was first computed in [DL06]. This is the case of the nine-variable polynomial that is the general 3 × 3 determinant.…”