Talk:Polynomial

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 $R[X]$" is (obviously) unfinished.

--Sébastien Moulin (talk me) 10:47, 1 April 2007 (CDT)

I have a question: From your description of the ring $R[X]$ it appears that you have in mind the ring of formal power series, which I would usually write $R[[x]]$. 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)

Brief comments

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)