Talk:Polynomial

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reaching the non-mathematical reader

I made a few changes in the lede, trying to make it transparently clear to the non-mathematical reader. Rick Norwood (talk) 14:15, 9 January 2010 (UTC)

• Thank you for doing it that way. Now, it is using "the most important" phrase again. Thats one of the points that I wanted to change. You can do it if you want.  franklin  14:33, 9 January 2010 (UTC)

Since I've been critical of earlier edits, let me be quick to say that this edit: Current revision as of 19:01, 9 January 2010 seems to me a good one. Also, I agree with  franklin  that whether or not an "expression" is finite depends on the context, and so the inclusion of the word "finite" is important. For example, in certain contexts an infinite sum is called an "expression". Rick Norwood (talk) 21:21, 9 January 2010 (UTC)

Here I disagree. A formula always refers to a finite collection of symbols, otherwise its cannot be written down. It would be pedantic to say so in the articles formula or expression, since everything discussed there is finite from the outset. It would be equally pedantic to say that a book must be made up of a finite number of pages. Infinite summations can be defined, and written down in a (finite) formula like

$\sum _{i=0}^{\infty }2^{-i}$,

but that formula does not involve arithmetic operations only; it involves a infinite summation. An infinite summation cannot be defined in terms of addition only, it also involves the notion of limit (and therefore some form of topology and a consideration of convergence). The mentioned expression article contains an example with a summation operator (with finite range though) which it correctly does not call an addition; the two are distinct (though related) notions. As far as I can tell, articles like formal power series, power series and series (mathematics) avoid ever calling these objects expressions or formulas.

Personally I disagree with using "finite" in the initial sentence of this article, for the same reason. However given the place in the aricle its presence may be justified by the fact that it may be helpful to people who might have seen things like

$1+x+x^{2}+x^{3}+\cdots$,

and believe that is an infinite expression (while it is just a sloppy way of writing down an infinite summation without invoking the corresponding operator).

Back to the "solving polynomial equations" section. Even if it weren't the case generally, I think it is perfectly clear that all formulas mentioned there are finite. An infinite formula for solving exactly an equation would not be of much use, even if one overlooks the impossibility of writing it down. And the statement needs a formalistic interpretation of what "formula" means, in order to be true; even with "finite" in place it is easy to write the tautological formula

$\{\,x\mid ax^{5}+bx^{4}+cx^{3}+dx^{2}+ex+f=0\,\}$

giving the solutions of a general quintic equation, and claim that it involves only a finite number of arithmetic operations. Also, in spite of what Franklin tells me, there is no mention of infinite formulas in the remainder of the section (or the article for that matter); hypergeometric, Siegel and theta functions are (additional) functions, not formulas involving arithmetic operations and radicals.

I'll close with a word for Franklin. I've looked at your recent edits, and see that they are made in good faith, and do contain improvements. I also note that your command of the English language is imperfect, no doubt it is not your mother tongue (take no offense, for me it is the same). Also you are visibly not acquainted with the subtle use of language in mathematics, which distinguishes "Terms of multivariate polynomials cannot be totally ordered by degree" (true) from "Terms of multivariate polynomials cannot be totally ordered" (manifestly false; every mathematician will note that writing down a polynomial involves totally ordering its terms) and "Terms of multivariate polynomials cannot be naturally ordered by degree" (a confused statement, since "naturally is undefined, and the sentence holds without it anyway). Also look up the meaning of "weasel words" in the context of Wikipedia, and you'll find that you are constantly using it for things that are not. I do approve of your attempts to make the language more suited for Wikipedia though. But it would be wise to start making edits that just replace formulations by objectively better ones, without making edits that depend more on your personal point of view. That will improve your credit with those following the changes to this article, and avoid to possibility that in some weeks/months time somebody takes issue with some result of your edits, dives into the history and finds that you have been making many edits that are disputable, and ends up with a major undo of all of them including those that were objective improvements. Once you have completed those improvements, you can start making more delicate changes, but do accept that people might disagree and revert some of them (without touching the previous ones). No one person should determine what goes in an article, and everybody should accept that. If you dislike being reverted, you might do better to put your effort into other articles. I know from experience that the Polynomial article serves a very varied public, and it is therefore very difficult to hit the right tone both for the very naive readers and for those that insist on mathematically precise and correct formulations. I have been reverted on major editing of this article that I thought (at the time) were improvements, and I've since decided to settle for putting my creative effort into more specialised articles, while only looking out form manifestly wrong formulations here. Marc van Leeuwen (talk) 11:38, 10 January 2010 (UTC)

There is a difference between a "formula", especially a "well formed formula", and an "expression". The latter word is more general. In any case, it does no harm to include the word "finite", if only to insure that Calc II students understand that a Mclauren Series is not a polynomial, and that in the phrase "infinite polynomial", "infinite" is not an adjective.

On the other hand, I agree that "total order" is not what we want in the discussion of the order of terms in a polynomial. I was going to work on that sentence today. Rick Norwood (talk) 13:26, 10 January 2010 (UTC)

• OK, but keep in mind blaming degree as the ultimate culprit of the problem with multivariate polynomials and not the polynomials themselves.  franklin  14:35, 10 January 2010 (UTC)

• I am happy with the new Classification section. It is the other option that I mensioned before: Not to compare univariate and multivariate polynomials according to the possibility of totally ordering the terms. Wikipedia is a very influential website, and it is our responsibility to avoid spreading misconception around as it was the section at the beginning. Now it doesn't really fight it since it doesn't talk about it at all, but that is fine too.  franklin  14:44, 10 January 2010 (UTC)

• I don't know why you said that I don't distinguish between: "Terms of multivariate polynomials cannot be totally ordered by degree", "Terms of multivariate polynomials cannot be naturally ordered by degree" and "Terms of multivariate polynomials cannot be totally ordered". When I precisely started editing that section because it was saying "Univariate polynomials have many properties not shared by multivariate polynomials. For instance, the terms of a univariate polynomial are completely ordered by their degree, and it is conventional to always write them in order of decreasing degree." A statement that doesn't put enough strength in emphasizing that is because of using the degree that the terms are not being totally ordered. And we saw lively, in the course of this discussion, what common mistakes occur when that is not explicitly said (the a^2+A^2+alpha^2 phenomenon). Let me add(this was written afterwards): Notice, in the original state of the article it says that univariate and multivariate are different and then says that univariate can be ordered by degree. This non-English speaker and non-acquainted with the subtle use of language in mathematics, can notice that that phrase have space for asking: "OK, the univariate are ordered by degree, but what about multivariate? It never said. Can they be ordered? Can they be ordered by degree?" And that's the origin of the problem. Am I misunderstanding that the writing was very poorly done there? Independently of the actual language, that is just clumsiness. A comparison is being established and then only elements of one of the compared items were being given.

About finite and infinite. Thanks for leaving the word finite around. Let's talk about philosophy since it is what is emerging now into the conversation. What is infinite? You make a distinction between the possible definition of $\sum_{k=0}^{\infty}$ and infinite. Then, to better understand what is happening, let's ask what is infinite? If we look at any other occurrence, the same thing is going to happen and we will end up not calling infinite to anything. You said "A formula always refers to a finite collection of symbols, otherwise its cannot be written down." and in this you are not quite right. Again, what do we call infinite? Even in logic, where the overwhelming number of times it is used to mean a finite thing, infinite formulas are studied sometimes. Again it is a matter of what to call infinite. Every time something is infinite it can be redefined (or rewrited, or re-thought) as finite operations with an extra (new) operation. An infinite sum, an integral, a limit that is infinite, a cardinal, a dimension, all of them can use a different wording that doesn't use the Platonic meaning of infinite. Following your ideas you will end up not calling infinite to anything, but that is not quite common. The distinction in that place of the article is to notice what is the main difference between the theorem of non-existence of formulas and the existence of the hypergeometric formulas. When you said that finiteness in implicit in the notion of formula you are being wrong since the hypergeometric ones are also formulas, they give a recipe to compute the solutions in every case. On the other hand you are right in saying that it would be implicit when saying that we want only arithmetic operations, but it is still too hidden for the general reader. I wouldn't be sure how to explain to someone how an absolute convergent series does not involve only the arithmetic operation of sum. Notice that saying that there is a limit and saying that there is an infinite sum makes no distinction since it is just a matter of language. There is no actual phenomenon making a difference.

About not liking to be reverted. I was complaining because the revertions were done to change a point and were changing many other places as well. I don't mind that in a few days those things get changed, that's how Wikipedia work. If I am around I will just check that the changes preserve what is being attained. For instance you changed the wording in the "Solving ..." section. I guess you wanted to say the last word. But preserved the essence of what I wanted to be included. I am still happy with that.  franklin  14:32, 10 January 2010 (UTC)

Definition II

Shouldn't we consider a polynomial being infinite in length? Just look at Taylor's theorem thrown onto the Complex plane. Here's a theorem: Suppose that a function f is analytic throughout a disk |z-z₀ < R₀ centered at z₀ and with radius R₀. Then f(z) has the power series representation

(1) f(z) = $\sum _{n=0}^{\infty }$a_n(z-z_0)ⁿ

where

(2) a_n = ${\frac {f^{(n)}(z_{0})}{n!}}$

That is, series (1) converges to f(z) when z lies in the stated open disk.

In other words, this is the expansion of f into a Taylor series about z₀. This can be an n^th order polynomial, so that means the polynomial is not finite; it is infinite in this case. Therefore, the first line should be revised to something like, "In mathematics, a polynomial is an expression of either infinite or finite length..."

This theorem came directly from a book called: Complex Variables and Applications, 8th edition, James W. Brown, Ruel V. Churchill, McGraw Hill 75.140.155.196 (talk) 17:52, 25 February 2010 (UTC)

A polynomial-like expression that has an infinite number of terms is called a power series. Mathematicians reserve the term "polynomial" for the finite case. Gandalf61 (talk) 14:03, 26 February 2010 (UTC)

Hiatuses

Parts of the second sentence seem to be missing: "For example, is a polynomial, but is not, because its second term involves division by the variable x (4/x) and because its third term contains an exponent that is not a whole number (3/2)." Fwend (talk) 17:05, 21 May 2011 (UTC)

If you are not seeing the formulas, your browser must be having a problem with fonts. Do you see anything between the quotes here: "x" and here "x + y × z²" and here "x + y × z²"? Marc van Leeuwen (talk) 19:09, 21 May 2011 (UTC)

I can see the content between the first 2 quotes, but not the last. Strange. I'm Using FF 4.01 for win32. It works fine in IE (version 8)

Etymology and history

Following several edits on the etymology, it would be interesting to add in the history section the name of the mathematician who introduced the term of polynomial and the language in which this term was introduced. By the way, there is something strange in the etymology of the English word: the suffix "ial", which is usually only for adjectives, is used also for substantive form (a polynomial). This is specific of English. In French we have polynôme for the substantive and polynomial for the adjective. D.Lazard (talk) 17:27, 5 June 2011 (UTC)

Use of the word "polynomial" by MS Excel spreadsheet

Microsoft Excel spreadsheet uses the word 'polynomial' to name a particular type of trendline it can fit through points on a chart. The equation it can return on the chart after the curve has been drawn will usually contain non-integer factors of x (as in y = 0.118x2 +0.283x + 3.055). This Wikipedia article, if I read it correctly, is saying that strictly these are not polynomials because they contain non-integers. Since I am not trained in this field, I genuninely don't know: that is why I was referring to the article. But I am surprised, and now confused. So can one of the contributors perhaps just include a note about this use of the word 'polynomial' by Excel, and include a note on whether such non-integers can validly be called polynomials either mathematically and/or in everyday parlance. Also, if necessary, refer the reader to the correct name for this type of equation, if it is not actually a polynomial. If I have it all wrong, so that it is simply a matter of clarifying for the non-expert, then could that be done? Ta. Stringybark (talk) 21:52, 2 April 2012 (UTC)

You have misunderstood what our article says. The exponents (or "powers") in a polynomial have to be integers, but the coefficients (the multipliers of the powers of x) do not have to be integers. So

$y=0.118x^{2}+0.283x+3.055$

is a perfectly good polynomial. Gandalf61 (talk) 09:02, 3 April 2012 (UTC)

I made a subtle tweak to the example in the lead, in order to show a case where non-integer coefficients are used. - DVdm (talk) 13:38, 3 April 2012 (UTC)

Nevertheless, IMO, the lead needs further edits to avoid such confusions: Strictly speaking the given definition is not that of a polynomial, but of a polynomial expression, i.e. an expression which may be rewritten, using the properties of the operations, into a polynomial ($(x-y)(x+y)$ is not a polynomial, even if, as an element of a polynomial ring, it is equal to the polynomial $x^{2}-y^{2}$. They are not equal as expressions.). The right definition of a polynomial is, IMO, "a finite sum of monomials, a monomial being, ...". Moreover to be not too technical, the univariate case should be considered first and separately. I will not immediately edit the lead in this direction because I do not see clearly how to do this, keeping a sufficiently low level of technicality. Help would be welcome. D.Lazard (talk) 14:46, 3 April 2012 (UTC)

Yes, it's always difficult to find the perfect an acceptable balance between accessibility and precision, specially in the lead. - DVdm (talk) 16:04, 3 April 2012 (UTC)

My mistake initially was to read the lead too hastily. Mea culpa. But I think the tweak is beneficial nonetheless. Thanks.Stringybark (talk) 12:45, 4 April 2012 (UTC)

I for one would like to see an explanatory link for just what x2 − 4/x + 7x3/2 is if it is not a polynomial.--24.212.154.2 (talk) 09:28, 7 February 2013 (UTC)

Properties of the roots

I would suggest to remove the paragraph about polynomial#Properties of the roots. It is not only technical but also completly unrelevant for this article. The only thing I would have expected under this headline would be Vieta's formulas and this with no more than the case for say second order polynomial and a link. What do other users think? --Falktan (talk) 19:17, 17 July 2012 (UTC)

At the very least the section should be named "Statistical properties of the roots". But I agree that this material treats a highly specialised question, and its inclusion at this point seems unjustified. Marc van Leeuwen (talk) 02:55, 18 July 2012 (UTC)

I agree with the two above comments. However this is an important question that deserves to appear in WP and to be expanded (the fact that the mean number of real roots is around the logarithm of the degree is lacking). However, for the moment, I do not see a better place. I suggest first to transform this section into a subsection of the preceding one, and to rename it "Statistical repartition of the roots". In a second step, the best is to create a new article Polynomial equation that contains the material of these section and subsection, together with some part of Root finding, with summaries and hatnote "main" in these two articles. D.Lazard (talk) 12:25, 18 July 2012 (UTC)

I moved this content (see copy box at the beginning of this talk page) --Falktan (talk) 20:16, 18 July 2012 (UTC)