Definition talk:Polynomial over Ring

I know the last two sections (polynomial functions and something else) don't quite fit with the previous definitions. I'll add a page "Evaluation Homomorphism" which will cover it.

Also, it's hard to say whether to group the definitions as I have done. I think this way is better so I'll make a page for each def. and redirect it.


 * Bear in mind we already have Evaluation Isomorphism which may or may not have common points of reference. --prime mover 00:24, 9 February 2011 (CST)


 * I wrote up a new page just for reference, looks like Evaluation Isomorphism should cover it though, I'll read it carefully soon.

Noncommutativity
I propose the following convention:


 * Polynomial means polynomial over a commutative ring with identity


 * Polynomial over a noncommutative ring is spelt out in full if/when it's needed.

There's no nice relation between polynomial functions and forms in the noncommutative case, so I think it's better to separate it out. --User:linus44

refactoring needed
We need a rethink of the way polynomials are defined on this site. The current way of presenting it is too abstract too early, particularly on Definition:Polynomial/Polynomial Form. Also the page Ring of Polynomial Forms introduces the notation of the form $\Z[X]$ etc. which is then used as a general notation used to specify a polynomial which, on the pages it is used, is not linked to and therefore taken for granted.

The $\Z[X]$ etc. ($\Q[X]$, $\R[X]$, whatever) notation really needs to be introduced on the Definition:Polynomial page if at all possible, allowing $f \in \Q[X]$ etc. to be dropped into any page discussing polymials without necessitating a load of explanatory text.

I'm prepared to take this one on, but it's going to be a tricky job to do properly and I may have to backtrack a few times. I am also unable to coherently put into words exactly what I believe needs to be done, specifically, to enhance this area. I got bogged down when I was doing Hartley and Hawkes, whose source citations seem to have vanished without trace, and Linus44 was simultaneously working from Grillet and restructuring it as he went.

What I will probably end up doing is returning to Hartley and Hawkes, and reprocessing it with Clapham, Binmore and Warner as backup. As I do not have the Grillet, I will need to leave references to that in a SourceReview template.

I don't know if I can get to this in the immediate future - it just cropped up in my awareness with the recent attention to Definition:Content of Polynomial. --prime mover (talk) 12:11, 9 March 2013 (UTC)


 * I agree that the current set-up on the polynomial section is too abstract. OTOH we do want to give a complete and rigorous treatment; this will probably require the elementary page Definition:Polynomial to be quite large and filled with references to other large pages that treat it in full rigour cq. full generality. It is critical that familiar (and highly important) cases like $\Z[X]$ and $\Q[X]$ are introduced on the basic page to "comfort" casual readers.


 * Admittedly this all is not an easy task and I'm happy to review the progress on it; I thank PM in advance for his efforts on the matter. &mdash; Lord_Farin (talk) 15:08, 9 March 2013 (UTC)