A formula for the Mahler measure of \(axy+bx+cy+d\)
Smyth, Boyd, Rodriguez Villegas and others found several explicit expressions for the logarithmic Mahler measure \(m(P)\) of polynomials in several (mainly in two) variables. In some cases, deep tools lying far from classical analysis were used to prove such equalities although numerically they can be verified without difficulty. In this paper, the author finds such an explicit expression for the Mahler measure of \(P(x,y)=axy+bx+cy+d,\) where \(a,b,c\) and \(d\) are certain complex numbers. Assume that they are all different from zero and satisfy the condition \(ad \neq bc.\) (The Mahler measure in all degenerate cases is also computed.) The answer depends on whether there will be zero, one, two or infinitely many complex pairs \((x,y)\) satisfying \(|x|=|y|=1\) and \(P(x,y)=0.\) The most difficult is the case with two such pairs. The answer is then expressed in terms of the coordinates of these two pairs and via the Bloch-Wigner dilogarithm function \(D(z)\), which is defined as \(\log |z|\arg (1-z)\) plus the imaginary part of the dilogarithm \(\sum_{k=1}^{\infty} {z^k \over k^2}\) for \(z \in {\mathbb C} \setminus [1, \infty).\) In all other cases (zero, one or infinitely many intersection points), it is shown that \[ m(axy+bx+cy+d)= \max \{\log |a|, \log |b|, \log |c|, \log |d|\}. \] The examples which evaluate \(m(P)\) for \(P(x,y)=2xy-3ix+3y+4,\) \(P(x,y)=ixy+x+y+1\) and those special cases which recover some earlier results, including the best known result on the subject \(m(x+y+z+1)=7\zeta(3)/2\pi^2,\) are also given.
scientific article; zbMATH DE number 1934763
scientific article