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 Definition: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 Definition: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

I agree. Polynomials over rings without unity are somewhat pathological as well, since we tend to say $X\in R[X]$ is a polynomial. --barto (talk) 05:04, 18 March 2017 (EDT)

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 Form. Also the page Definition: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. — Lord_Farin (talk) 15:08, 9 March 2013 (UTC)

1965: Serge Lang: Algebra (think he works with sequences in 1987: Serge Lang: Undergraduate Algebra) follows the same construction as Grillet; but in finitely many variables. It's slightly lacking in rigour, but should be more accessible. If i remember correctly; Bourbaki also construct polynomials this way.

A possibly better approach is to work with sequences (possibly see 1987: Serge Lang: Undergraduate Algebra, my memory is vague). i.e. a polynomial $\sum^n a_i X^i$ identifies with $(a_1,\ldots,a_n,0,0,\ldots)$. This would be more elementary but still rigorous; I prefered the other construction since it more fitting with the universal property for the polynomial ring. It's not majorly awkward with sequences. --Linus44 (talk) 22:37, 9 March 2013 (UTC)

We had that up in Feb 2011 but it got deleted. --prime mover (talk) 21:51, 12 March 2013 (UTC)

That's good; hopefully just restoring that will save some work. How about the following:

Polynomial has three definitions. One is the "naive" one (it's still rigorous, but get's difficult to work with in some situations), say $A \subseteq B$ rings, $x \in B$ then a polynomial is $a_0 + a_1 x + \cdots + a_n x^n$, $a_i \in A$. This was up before as well.

Then the sequence definition, used for finitely many variables since it's a good compromise in this situation.

Then finally the existing definition, since this makes far more sense for polynomial rings with infinitely many variables.

It should be possible to save having three proofs of everything with a couple of well thought out lemmas. Good plan? (not that I'll have time to do anything on it...) --Linus44 (talk) 23:00, 12 March 2013 (UTC)

Probably. I'm taking my time over it. Then there's the "basic" definition in which the rings in question are $\R$ and $\Q$ and $\Z$ etc. I want to make it so that someone stumbling over a "polynomial" in Analysis (real and complex) don't have to digest the concepts of "ring" and "integral domain" before they can get a handle - especially when the jargon like "monic" and "leading coefficient" and "degree" emerge. --prime mover (talk) 23:11, 12 March 2013 (UTC)

I don't have access to books I haven't got access to (which admission made a contributor cross a few weeks ago who declared I had no right to be allowed to contribute to this site without having first studied the books he had) so afraid I can't put the Lang up till I've restored the above balance and that won't happen tonight soz. The sequence approach is done in one of my books and I that I'd already put it up, maybe it's one of the pages that got deleted in the purge 2 years ago. At the time I did not think I was allowed to argue for providing all the definitions, I thought we had to stick to only one, so I allowed all that to get deleted. I will go and look for it again. But I'm tired and I don't know whether I'm going to be able to do all this when I'm not tired because I'm too tired to tell. (What I really want is for the entire universe to vanish now, forever, and never ever come back again but I expect that's beyond my control.) --prime mover (talk) 22:48, 9 March 2013 (UTC)

I didn't mean that as a demand you get to work on it; it was more of a "working on the page structure is complicated, so I'll expound some general philosophy on the talk page"-type comment. --Linus44 (talk) 23:04, 9 March 2013 (UTC)

PS: I reserve the right to use profanities when I want. There is precedent.

Inconsistency

Integral domain def has element, others have expression. Aside from that, these look like several identical definitions except over different sorts of rings. What is the difference justifying the distinction? --Dfeuer (talk) 02:36, 28 March 2013 (UTC)

Okay, you do it. It's a difficult area to get right. I've obviously done an inadequate job, I'll leave it for an expert to do. --prime mover (talk) 06:31, 28 March 2013 (UTC)

Bundle polynomial "expressions" in a separate subpage

There are three weird definitions of things called "expressions":

Definition:Polynomial (Abstract Algebra)/Ring

Definition:Polynomial (Abstract Algebra)/Integral Domain

Definition:Polynomial (Abstract Algebra)/Field

I suggest to move them to a separate (sub)page, because they don't help to understand what a polynomial actually is. They only reinforce the confusion between polynomials and polynomial functions. --barto (talk) 19:34, 15 March 2017 (EDT)

I haven't got a slightest clue what to do with this area. I had a vague start at defining a polynomial, then a contributor came along and completely changed it to the most abstract and general form, and made it difficult to reconcile with the standard, elementary, intuitive notion of what a polynomial is.

If you have a handle on this, please go ahead and rip it to pieces and rebuild it. --prime mover (talk) 19:40, 15 March 2017 (EDT)

...where "rip it to pieces" and "rebuild it" are not two separate things. Ripping apart is easy, also rebuilding with some bigger picture in mind is where the challenge lies. — Lord_Farin (talk) 16:08, 16 March 2017 (EDT)

Yes, I'm aware that it's not an easy task. Of course I'd keep definitions that require different levels, the first actual formal definition being probably the one with sequences. A bigger picture could consist of elaborating on the isomorphisms between them, and why it's not such a problem to have definitions that do not, formally, coincide. Things like $R[X,Y]\cong R[X][Y]$. --barto (talk) 16:53, 16 March 2017 (EDT)

FYI, those pages are now at Definition:Polynomial in Ring Element. --barto (talk) (contribs) 03:11, 15 January 2018 (EST)

Explaining the notation R[X]

I have made sure to explain the notation $R[X]$ here, for the convenience of being able to write Let $f \in R[X]$ be a polynomial, with polynomial linking here. It is the essence of this page. --barto (talk) (contribs) 18:22, 15 January 2018 (EST)

Good start. --prime mover (talk) 18:30, 15 January 2018 (EST)