We study a class of two-variable polynomials called exact polynomials which contains $A$-polynomials of knot complements. The Mahler measure of these polynomials can be computed in terms of a volume function defined on the vanishing set of the polynomial. We prove that the local extrema of the volume function are on the two-dimensional torus and give a closed formula for the Mahler measure in terms of these extremal values. This formula shows that the Mahler measure of an irreducible and exact polynomial divided by $\pi$ is greater than the amplitude of the volume function. We also prove a K-theoretic criterion for a polynomial to be a factor of an $A$-polynomial and give a topological interpretation of its Mahler measure.

我们研究了一类二变量多项式，称为精确多项式，其中包含$ A $ -结补数的多项式。这些多项式的马勒测度可以根据在多项式消失集上定义的体积函数来计算。我们证明了体积函数的局部极值在二维圆环上，并根据这些极值给出了马勒测度的封闭公式。该公式表明，不可约和精确多项式除以$ \ pi $的马勒测度大于体积函数的振幅。我们还证明了多项式的K理论准则是$ A $多项式的因数，并给出了其Mahler测度的拓扑解释。