The Mahler measure of exact polynomials and special $L$-values of $K3$ surfaces

We express the Mahler measure of an exact polynomial in arbitrarily many variables in terms of Deligne-Beilinson cohomology. We then focus on the relationship between the Mahler measure of four-variable exact polynomials and the special value of the $L$-function of $K3$ surfaces at $s=4$. This result extends the three-variable case studied in \cite{Tri23}. Finally, we prove, under Beilinson’s conjecture, that the Mahler measure of the polynomial $(x+1)(y+1)(z+1)+t$ is expressed in terms of the Riemann zeta function and the $L$-function of the modular form of weight 3 and level 7.