Question

There seem to be two conflicting definitions for p-adic valuation in the literature.

Firstly, for any non-zero integer n, we have $\nu=\nu_p(n)$ is the greatest non-negative integer such that $p^\nu$ divides $n$. Secondly, we have $|n|_p$ which is defined as $1/p^\nu$. [These definitions can be extended to the rationals.]

$\nu$ is defined as the p-adic valuation in Khrennikov, Nilson, P-adic deterministic and random dynamics (for example) and $|\cdot|_p$ is defined as the p-adic valuation in Khrennikov, P-adic and group valued probabilities, in Harmonic, wavelet and p-adic analysis (for example).

Question: Is there a preferred definition for p-adic valuation?

Answer 1

I will explain what's going on. We call $|x|_p$ the $p$-adic absolute value of $x$ and $v_p(x)$ the $p$-adic valuation of $x$. The distinction that is made by the two terms "absolute value" and "valuation" is completely standard... in English. However, Khrennikov is originally from Russia and in Russian there is one term for both concepts (нормирование = normirovanie, with stress -- for English speakers I am not making this up -- on the second syllable). There is a term "absolute value" in Russian, but it is not an abstract concept; it refers only to the usual absolute value on the real or complex numbers (and quaternions?). This is perhaps why Khrennikov is using the term "valuation" incorrectly to refer to an absolute value function.

(I'm giving a course in Moscow this semester and I found this point frustrating when I was preparing my initial lectures. In different books I found the same word used for an absolute value and for a valuation and couldn't find the term that exclusively means absolute value. Eventually I determined there isn't one; you just know by context what meaning is intended. Native speakers are welcome to correct me here.)

Answer 2

I would say it's incorrect to call $|\cdot|_p$ a valuation; it is an absolute value or better yet a norm, but certainly not a valuation. Note also that for any $0 < \alpha < 1$, the formula $|n|=\alpha^{\nu_p(n)}$ defines a norm and it's customary but by no means obligatory to take $\alpha = 1/p$.

See http://en.wikipedia.org/wiki/Valuation_%28algebra%29 for the definition of a valuation and the remark that "Some authors use the term exponential valuation rather than "valuation". In this case the term "valuation" means "absolute value"."

Answer 3

The conflict is just that some people use the words valuation and absolute value interchangeably. The term "p-adic valuation", used correctly, refers to $\nu$, though perhaps in some areas of math the prevailing choice is the other way around.