The alternative to Mahler measure of a multivariate polynomial

Abstract

We introduce the ratio of the number of roots of a polynomial $P_d$, less than one in modulus, to its degree $d$ as an alternative to Mahler measure. We investigate some properties of the alternative. We generalise this definition for a polynomial in several variables using Cauchy's argument principle. If a polynomial in two variables do not vanish on the torus we prove the theorem for the alternative which is analogous to the Boyd-Lawton limit formula for Mahler measure. We determined the exact value of the alternative of $1+x+y$ and $1+x+y+z$. Numerical calculations suggest a conjecture about the exact value of the alternative of such polynomials having more than three variables.

10 pages, 1 figure