Creation of the article
I just created this article and it is far from being finished. First, feel free to correct spelling or style (I am not so used to write in English). In case you are not sure about such a correction (for instance if you are not familiar with mathematics), just drop me a message on my talk page.
The article may evolve a lot in the future, I just tried to be bold writing this! I tried to write lively prose, not "encyclopedese", but I am not sure how the general tone fits with the Citizendium's standards (for instance, is the use of "we" ok ?). I also tried to make the text readable for a non specialist as much as I could but I do not know if I succeeded. For the time being, the article just gives a very abstract definition of polynomials without showing why it is useful! Maybe this part about the construction of algebras of polynomials should form an article in itself and only be summarized in the main article (for later parts, like the arithmetics one, it is even more obvious than a devoted article is needed). But for now, I guess this is better than a mere red link anyway.
I plan to continue to work on this article, the whole structure may evolve a lot, and the paragraph entitled "The algebra " is (obviously) unfinished.
I have a question: From your description of the ring it appears that you have in mind the ring of formal power series, which I would usually write . iss that correct, or am I trying to read too much into an evolving article? Greg Woodhouse 09:18, 3 April 2007 (CDT)
- I said I planned to continue this work, but finally I won't, leaving the Citizendium for reasons unrelated to mathematics. I wrote this sketch about polynomials on impulse, and I am now dissatisfied with the result, which may put too much emphasis about the formal aspect of things. In short, it looks more like a boring Bourbaki-like lecture than like a vivid introduction to the subject. I realized that while writing and that is the reason why I stopped in the middle of a section. I wondered about placing a 'speedy delete' template on the article but it may not be an appropriate case for doing so. So I leave to you to decide if it is better to restart the article from scratch or to try to improve the current version. I don't mind about either choice. Regards, --Sébastien Moulin (talk me) 08:08, 19 April 2007 (CDT)
I won't go into as much depth with comments here as I have on the discussion pages of Complex number and Prime number, since the article might be reworked soon; but the spirit of my comments are the same here as there.
When we teach polynomials to mathematics students, we teach how to manipulate them (adding coefficient by coefficient, for example); the next important topic is probably how to find their roots, first algebraically (think quadratic formula) and later analytically (Newton's method). Only after all this would we introduce a formal definition of a polynomial as an infinite sequence with finite support over a ring. (Zuh? they would rightly say.)
Formal definitions are for us mathematicians to reassure ourselves that the objects we've been using all along are consistent with the foundations of mathematics. They don't convey intuition - in fact they often take away from intuition, until years later when much deeper exposure to mathematics lends a greater perspective.
So in short (if I haven't already passed that): the primary contents of this article should be concerned with how most laypeople use polynomials. The formal definition (Bourbaki-ish, I agree) should be either very near the end or omitted completely.
All that being said, the first paragraph is a pretty good stab at the right way to start the subject. The reason why polynomials are so nice is that any property that's true for constants and x itself, and is preserved under taking sums and products, is automatically true for all polynomials (continuity, for example).
I think "Finding roots of polynomials" would make for a good section in this article. Also I'd hold off on multiple variables until the main ideas for single-variable polynomials have been communicated. (--Greg Martin)
- I was just about to say the same thing, but with less eloquence. I'm going to start the big edit of stuff after the introduction, to break the ice. I was just looking at the Wikipedia page -- it doesn't ever seem to mention how to multiply polynomials!Barry R. Smith 17:24, 11 December 2008 (UTC)
I created an advanced page, with the intention of discussing polynomials over commutative unitary rings and linking to pages about non-commutative polynomials. As the convention on several other pages (see vector space and quadratic equation) of using real number scalars or coefficients, I moved the definition of polynomials and their operations, as well as the word ring and the discrepancy between polynomials and polynomial functions to the advanced page.
I don't see a need to explicitly say that a distinction between polynomial functions and polynomials must be made on the main page, as it doesn't cause any difficulty when real number coefficients are used. Of course, the relation between a polynomial and the associated function should be mentioned, but not the fact that there is a big conceptual difference between the two in certain circumstances. But that's just my opinion about the focus of the main page -- be bold and change what I did if you think it appropriate.Barry R. Smith 19:49, 22 December 2008 (UTC)
- By coincidence I was just thinking about an article for Polynomial ring which might take up some of the Advanced material. Richard Pinch 19:57, 22 December 2008 (UTC)
- Perhaps that would be better. Is there a policy against having "polynomial/advanced" duplicate "polynomial ring"?Barry R. Smith 20:02, 22 December 2008 (UTC)
- Looks good. You could change the "Noetherian domain" property to "commutative Noetherian ring", if you wanted. Do you intend to discuss polynomials in several variables there? How about linking to a page about free algebras? Perhaps "polynomial/advanced" can just cover polynomials over a field, since much of the "polynomial" page, when completed, would be almost identical in that context, and could be used by someone interested in polynomials over GF_2 who doesn't want the huge generality of the polynomial ring page.Barry R. Smith 00:45, 23 December 2008 (UTC)
The first sentence should be perfect, especially for an article as important as this. Taking Sebastian's invitation to change the style, I shortened the first sentence. It seemed rather lengthy, so I removed the part between the commas in "...variables, denoted by lower case letters like x, y, etc., ...". I also thought "etc." is awkward to have in the defining sentence, and in the second sentence it is made clear that the variables are being denoted by 'x', or 'x', 'y' etc.Barry R. Smith 20:05, 24 December 2008 (UTC)
"of one variable" vs. "in one variable"
The original article was written using the phrase "polynomial of one variable". This sounds strange to my (native American english) ear, as I am used to "polynomial in one variable" (although the former looks more sensible to my eyes -- but perhaps "in one variable" comes from the same places as "a play in three acts"). Googling both phrases supports my ear, so I went ahead and changed everything to "in one variable". However, if this is an issue of the Queen's english versus my American english, or some other purely cultural artifact, let me know and I'll be happy to change it back.Barry R. Smith 20:14, 24 December 2008 (UTC)
I just wanted to explain my drastic change to the introduction. In keeping with the Citizendium practice of making the intro explain the importance of the object of the page, I added information about applications of polynomials. On the other hand, examples of non-polynomials and a discussion of rational number or irrational coefficients did not seem, to me, pressing to the introduction of the topic and an explanation of its importance. In the interest of keeping the intro as brief as possible (to make a potential reader more prone to read through it), I moved this material to the first section on polynomials in one variable.Barry R. Smith 21:44, 24 December 2008 (UTC)