Identities of Mahler measures of certain polynomials defining curves of arbitrary genus. (English) Zbl 1467.11095

Wu, Xiaoliang; Liu, Hang; Tang, Guoping

J. Math. Anal. Appl. 494, No. 2, Article ID 124602, 12 p. (2021).

In this paper, the authors give some identities for Mahler measures of polynomials in two variables. For example, in Theorem 2, defining \(P_k(x,y)=y^2+(kx+1)y+x^3\), they show that \[2m(P_k(x-1/k,y)) = 3m(P_k(x,y))-\log |k|\] for \(|k| \geq k_0=3.2407\dots\), where \(k_0\) is the root of some quartic polynomial. Similar identity \[m(Q_k(x-1/k,y)) = 2m(Q_k(x,y))-\log |k|\] is obtained for the shift of the polynomial \(Q_k(x,y)=y^2+(2x^2+kx+1)y+x^4\) for \(|k| \geq k_1=5.9794\dots\), where \(k_1\) is the root of some quintic polynomial.

Reviewer: Artūras Dubickas (Vilnius)

MSC:

11R06	PV-numbers and generalizations; other special algebraic numbers; Mahler measure
11R09	Polynomials (irreducibility, etc.)
19F27	Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects)

Keywords:

Mahler measure; regulator integral; \( K_2\); special value of \(L\)-function

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