Non-Archimedean absolute value

Definition

A non-Archimedean absolute value on a field $K$ is a map

|\cdot|:K\longrightarrow\mathbb R_{\geq0}

such that

|x|=0\Longleftrightarrow x=0,\qquad |xy|=|x||y|,\qquad |x+y|\leq\max\{|x|,|y|\}.

The last condition is the strong triangle inequality.

It induces the ultrametric $d(x,y)=|x-y|$ . If $v:K\to\mathbb R\cup\{\infty\}$ is an additive valuation, then

|x|=e^{-v(x)}

is a non-Archimedean absolute value. Conversely, taking $-\log|x|$ recovers an additive real-valued valuation.

References

Oleg Viro, “Hyperfields for Tropical Geometry I: Hyperfields and dequantization,” 2010. arXiv:1006.3034.