Calculus on Fields - Heights of Polynomials, Mahler's Measure and Northcott's Theorem

Heights

Definition. For a polynomial with coefficients in a number field $K$

$$f(t_1,\dots,t_n)=\sum_{j_1,\dots,j_n}a_{j_1\dots j_n}t_1^{j_1}\dots t_n^{j_n}=\sum_{\mathbf{j}}a_{\mathbf{j}}\mathbf{t}^{\mathbf{j}},$$

the height of $f$ is defined to be

$$h(f)=\sum_{v \in M_K}\log|f|_v$$

where

$$|f|_v=\max_{\mathbf{j}}|a_{\mathbf{j}}|_v$$

is the Gauss norm for any place $v$.

Here, $M_K$ refers to the canonical set of non-equivalent places on $K$. See first four pages of this document for a reference.

As one can expect, this can tell us about some complexity of a polynomial, just like how the height of an algebraic number tells us its complexity. Let us compute some examples.

Computing Heights

Let us consider the simplest one

$$f(x)=x^2-1 \in \mathbb{Q}[x]$$

first. Since $|x^2-1|_v=1$ for all places $v$, the height of $f$ is a sum of $0$, which is still $0$.

Next, we take care of a polynomial that involves prime numbers

$$g(x)=\frac{1}{4}x^4+\frac{1}{3}x^3+\frac{1}{2}x^2+x+2$$

We see $|g(x)|_\infty=2$, $|g(x)|_2=2^{-(-2)}=4$, $|g(x)|_3=3^{-(-1)}=3$, and...