Short Mahler-measure-preserving multiples of multivariable polynomials

Abstract

The $2$-variable polynomial $(y + 1)x^2 + (y^2 + y + 1)x + (y^2 + y)$ has length $7$; it is a factor of the length-$6$ polynomial $(y+1)x^3 + x^2 - y^3x - y^3 - y^2$ which shares the same Mahler measure. On the other hand, consider the $2$-variable length-$7$ polynomial $(1+x) + (1-x^2+x^4)y + (x^3+x^4)y^2$. Extensive computations by Boyd and Mossinghoff suggested strongly that this has no length-$6$ multiple with the same measure. But how can one prove this? In this paper we develop a method which attempts to find the shortest multiple of a polynomial in $\mathbb{Z}[z_1,\dots,z_m]$ such that the multiple has the same Mahler measure as the original polynomial. The method is heuristic: it might fail (although we have yet to find an example when it does fail), but when it succeeds it provides a proof of shortness. In particular we can remove any doubt concerning the Boyd-Mossinghoff example mentioned above, and we are able to find shortest-possible Mahler-measure-preserving multiples of all the known $2$-dimensional examples having measure below 1.37