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I've been looking around and haven't found anything on this topic so I figured I'd ask here.

Let $Z(x)$ and $R(x)$ be the rational functions over the integers and real respectively.

Does there exist a norm on $Z(x)$ (or $R(x)$) such that there exist a $p\in Z(x)$ (or $R(x)$) so that for all $n\in N$ we have $||p^{n+1}||< ||p^{n}||$?

Furthermore assuming the norm we are looking for in the above paragraph does exist, can we then find a norm on the algebraic closure of $Z(x)$ (or $R(x)$) so that it agrees with the previous norm when the domain is restricted to the rational functions?

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Are you familiar with the $p$-adic norm on the integers, and its extension to the rationals, and its completion? You can impose a similar norm on the rational polynomials. – Gerry Myerson Mar 9 2011 at 1:40
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 What does "extend this norm, along with its property" mean? and what do you mean by its closure? The algebraic closure? As stated, the property just has to hold for at least one $p$, so will automatically hold in any extension. So, this question just does not seem clear to me. And, I expect that something like the $p$-adic norm will work if you do clarify this. Also, what's the motivation? I'm voting to close this question as it is. – George Lowther Mar 9 2011 at 2:46
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 Voting to close. OP seems unwilling to engage. – Gerry Myerson Mar 11 2011 at 5:02
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 The answer to your last question is "yes". – S. Carnahan♦ Mar 11 2011 at 7:44
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 "Discussion etiquette" on MO (or anywhere else) is not all that tricky. When someone says something that might be helpful, you express gratitude. Then you go to the library to try to figure out what the heck the person was saying. If you get stuck, you come back to report such progress as you've made, and ask what to do next. – Gerry Myerson Mar 11 2011 at 11:48
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closed as not a real question by George Lowther, Franz Lemmermeyer, Andres Caicedo, Gjergji Zaimi, Gerry Myerson Mar 11 2011 at 5:02

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