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arXiv · 2502.02803

The alternative to Mahler measure of a multivariate polynomial

Dragan Stankov
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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.

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Dragan Stankov. 2025-02-05. The alternative to Mahler measure of a multivariate polynomial. https://arxiv.org/abs/2502.02803

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