alternative definition of valuation

A valuation on a field $\mathbb{K}$ is a map $|\cdot|\colon\mathbb{K}\to\mathbb{R}$ such that

1. 1.

$|x|=0$ if an only if $x=0$

2. 2.

$|xy|=|x|\,|y|$

3. 3.

$|x+y|\leq C\max\{|x|,|y|\}$

The quantity $C$ which appears in the third criterion is a positive real number which is known as the .

There is some flexibility in the choice of the constant $C$ in this definition — one can replace $C$ by a larger number $C^{\prime}$. To deal with this ambiguity, one defines the of the valuation as

 $\inf\{C\mid(\forall x)(\forall y)\>|x+y|

The relation  of this definition to the usual one is the following. On the one hand, if $|\cdot|$ satisfies the usual definition, then

 $|x+y|\leq|x|+|y|\leq 2\max\{|x|,|y|\}$

so a valuation in the old sense is a valuation in the new sense with constant 2.

On the other hand, suppose that $|\cdot|$ satisfies the alternative definition with constant $C<2$. Then we have the following result.

If $|\cdot|$ is a valuation according to the definition of this entry with constant $C\leq 2$, then $|\cdot|$ satisfies the identity

 $|x+y|\leq|x|+|y|.$

The proof of this assertion is given in a supplement to this entry.

The foregoing discussion shows that the new definition is more general than the old definition precisely when $C>2$. However, this extra generalty is not as great as it might seem at first sight. As is obvious from examining the definition, if $|\cdot|$ is a valuation, then so is $|\cdot|^{p}$ for any power $p>0$. Furthermore, if the valuation has constant $C$, then valuation $|\cdot|^{p}$ has constant $C^{p}$. Therefore, given any valuation $|\cdot|$ in the sense of this entry, there will exist a number $p$ such that $|\cdot|^{p}$ is a valuation in the sense of the parent entry. Moreover, given the fact that two valuations which are powers of each other are equivalent     , one sees that the extra generality is not that interesting since the new valuations are equivalent to the old valuations.

Title alternative definition of valuation AlternativeDefinitionOfValuation 2013-03-22 14:55:47 2013-03-22 14:55:47 rspuzio (6075) rspuzio (6075) 7 rspuzio (6075) Definition msc 11R99 msc 12J20 msc 13A18 msc 13F30