Ok. My first remarks (it's my opinion, I can be wrong, I can be a dick, it's your call):
Why not introduce fundamental thm 3.8 into the introduction. It was not clear that the finite generators were those si functions, you define later. I know that it's not easy to explain it without a lot of details but...>Some remarks about the contents
f(g.A)=f(A) and not f(g.A)=A
* Definition 3.11
Aren't everyone aware that there is only one topology for a normed vector space of finite dimension? Why do you precise?
* Lemma 3.12
Is it necessary? It's a basic theorem about continuous extension of a function (I don't know how to say it in English).
* 4 the proof
The first introduction phrase, is not necessary. Just say that you'll use the following U. Don't need to say that it was a contest between a lot of candidates.
* Definition 4.2
Calling the set of companion matrix K confused me. K is your field. I made the confusion during Corollary 4.5, but you're coherent.
About the style
"We're dealing with"... I'm not sure, but I don't think that it's the kind of expression we want to read in a paper. English is not my first language, but I think that "to deal with" is too informal.>"tricks such as averaging?"
"Tricks"!!! Where do you think you are? In the ghetto? No, don't talk like that, the one reading your paper isn't your friend. I am not your pal, bro.>It is a standard result
That's your opinion. I don't think it's good practice to talk about your opinions.>It is no coincident two of these functions look familiar
It looks like you're animating a TV show. If you're talking to students, yes, you can say that. But a paper isn't a TV show.>require rudiments of field theory
It's your opinion, just state that it requires some results of field theory not cover by this paper. Of course, when talking you can say that.
By the way, thank for the paper, I knew that there was such a theorem, but that's the first time that I read it explicitly.