The alternative to Mahler measure of a multivariate polynomial

2025/02/05 by Dragan Stankov, Stankov, Dragan
Mathematics · #Functional Equations Stability Results #Advanced Statistical Methods and Models #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.2502.02803

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.

Related

• Some examples of Mahler measures as multiple polylogarithms
• Asymptotic expansion of the difference of two Mahler measures
• Mahler measure of P_d polynomials
• The Mahler measure of exact polynomials in three variables
• Mahler Measure of "Almost" Reciprocal Polynomials