Jackson \(q\)-Mahler measures (Q964227)

Let \(f\in{\mathbb C}[x_1,\ldots,x_n]\) and \[ S_f:=\{q\in (0,1)^n\mid f(e^{2\pi iq_1^{j_1}},\ldots,e^{2\pi iq_n^{j_n}})=0\,\text{for some}\,(j_1,\ldots,j_n)\in({\mathbb Z}_{>0})^n\}. \] For \(q=(q_1,\ldots,q_n)\in (0,1)^n\setminus S_f\) the authors define the Jackson \(q\)-Mahler measure of \(f\) by \[ m^q(f)=(1-q_1)\ldots(1-q_n)\sum_{j_1,\ldots,j_n=1}^\infty \log|f(e^{2\pi iq_1^{j_1}},\ldots,e^{2\pi iq_n^{j_n}})|q_1^{j_1-1}\ldots q_n^{j_n-1}. \] They prove: Theorem. Let \(f\in{\mathbb C}[x]\) with the Mahler measure \(m(f)\). We have that \[ \lim_{q\to 1, q\not\in S_f}m^q(f)=m(f). \]