The alternative to Mahler measure of a multivariate polynomial

TL;DR

The paper defines an alternative to Mahler measure, $c(P)$, as the probability that a randomly chosen zero of $P$ lies in the open unit disk, and extends the construction to multivariate polynomials via Cauchy’s argument principle. It proves a Boyd–Lawton-type limit for polynomials in two variables that do not vanish on the unit torus, and computes exact values $c(1+x+y)=1/3$ and $c(1+x+y+z)=1/4$, with a conjecture that $c(1+x_1+\cdots+x_k)=1/(k+1)$ for $k$ variables, supported by numerical evidence and related limits such as $\lim_{n_2\to\infty}\cdots\lim_{n_k\to\infty} c(1+x_1+x_1^{n_2}+\cdots+x_1^{n_k})=(k-1)/(k+1)$. The results reveal asymmetry under variable order and place this probabilistic invariant on a parallel track to Mahler measure, suggesting broader applications in the study of zeros on the unit torus. The framework yields exact small-variable values and conjectured generalizations with potential connections to special values of $L$-functions.

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.

Figures (1)

• Figure 1: The determination of $\varphi$ such that $e^{\varphi i }+a$ (it is located on the blue circle) is inside of the unit circle (the red circle). It follows that $\varphi \in [\pi -\arccos(|a|/2), \pi +\arccos(|a|/2)]$.

Theorems & Definitions (16)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 2.1
• Definition 2.2
• Lemma 2.3
• Corollary 2.4
• Theorem 2.5
• Lemma 2.6