The alternative to Mahler measure of a multivariate polynomial

TL;DR

This paper proposes an alternative to Mahler measure based on the ratio of roots inside the unit circle, explores its properties, generalizes it to multivariate polynomials, and provides exact values and conjectures.

Contribution

It introduces a new root-based measure as an alternative to Mahler measure, extending it to multivariate polynomials and establishing foundational properties and conjectures.

Findings

Exact value of the alternative for 1+x+y and 1+x+y+z.

The alternative satisfies a Boyd-Lawton type limit formula.

Numerical evidence and conjectures for polynomials with more than three variables.

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.