The Mahler measure of a genus 3 family (Q829655)

In the paper under review, the authors show that the two polynomials \[ P_{k}(x,y):=y^{2}+(x^{6}+kx^{5}-x^{4}+2(1-k)x^{3}-x^{2}+kx+1)y+x^{6} \] and \[ Q_{k}(x,y):=xy^{2}+(kx-1)y-x^{2}+x, \] where \(k\in \mathbb{N}\) and \(k\geq 2\), have the same Mahler measure. The method of proof of this conjectural identity of [\textit{H. Liu} and \textit{H. Qin}, ``Data for Mahler measure of polynomials defining genus 2 and 3 curves'', \url{https://github. com/liuhangsnnu/mahler-measure-of-genus-2-and-3-curves}] is similar to that employed in [\textit{M. Lalín} and \textit{G. Wu}, Int. J. Number Theory 15, No. 5, 945--967 (2019; Zbl 1448.11191)], establishing identities between the regulators, but new strategies are used to simplify the regulator of the genus \(3\) curve \(P_{k}(x,y)=0\), when \(k\geq 3\), before comparing it to the regulator of the curve \(Q_{k}(x,y)=0\) of genus \(1\).