---
tag: 1PJS951C8N
kind: definition
name: Mahler measure
msc: [11R06, 11C08]
defines: QVPYBPRHRC
uses: [09FTAET5Z4, 0Q9EZS90GY, 00EHX6FWC0, 0JP0W07Y9A]
sources:
  - url: https://en.wikipedia.org/wiki/Mahler_measure
---

## Definition

Let $f \in \C[x]$ be a nonzero polynomial.
By the [[fundamental theorem of algebra|0JP0W07Y9A]], write $f = a (x - \alpha_1) \cdots (x - \alpha_n)$ with $a \in \C \setminus \set{0}$ and [[roots|00EHX6FWC0]] $\alpha_1, \dots, \alpha_n \in \C$.
The Mahler measure of $f$ is
$$M(f) = \abs{a} \prod_{i=1}^{n} \max(1, \abs{\alpha_i}).$$

## Remarks

By Jensen's formula, $M(f) = \exp\left( \int_0^1 \log \abs{f(e^{2 \pi i \theta})} \, d\theta \right)$.
It is multiplicative: $M(f g) = M(f) M(g)$.
Example: $f = 2x - 1 = 2(x - \frac{1}{2})$ gives $M(f) = 2 \cdot \max(1, \frac{1}{2}) = 2$.
Non-example of a common confusion: $M(f)$ is not the largest absolute value of a coefficient; for $f = (x + 1)^2 = x^2 + 2x + 1$ that number is $2$, while $M(f) = 1$.
If $f \in \Z[x]$ is nonzero, then $M(f) \ge 1$.
By a theorem of Kronecker, a nonzero $f \in \Z[x]$ has $M(f) = 1$ if and only if $f$ is $\pm$ a product of a power of $x$ and [[cyclotomic polynomials|08KEYFAH66]].

## Explanations

### L1 human

The Mahler measure looks at where the roots are.
Roots inside the unit circle are ignored, and roots outside count with their size.