Talk:Polynomial/Archive 4

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
 * $$\sum_{i=0}^\infty2^{-i}$$,

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) = $$\sum_{n=0}^{\infty}$$an(z-z0)n

where

(2) an = $$\frac{f^{(n)}(z_0)}{n!}$$

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 nth 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 "$x + y × z^{2}$" 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)

Properties of the roots
I would suggest to remove the paragraph about polynomial. 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)

Why?
When I go to the Automobile page, I get the 'Why' in the first sentence. 'Ah, it's used for transporting passengers!' With this polynomial page, I read it all and was still left with no understanding of why this definition exists. I don't like accepting rules without knowing the rationale and purpose behind them at the same time. I had to visit the second search result (http://www.mathsisfun.com/algebra/polynomials.html), and in a minute had a much clearer overview of polynomials, and near the end finally found a summary of the value of polynomials: 'Because of the strict definition, polynomials are easy to work with', calculations with polynomials result in polynomials, and they are easy to graph.'

This may be a lazy and incorrect summary, but to my currently non-mathematical mind it helped me understand why a definition attached to a seemingly-arbirtrary set of rules exists. I suppose the "Elementary properties of polynomials' implicitly touches on their value, but that would go over most non-mathematicians heads.

I'd also re-iterate that that secondary link was far more accessible to the non-mathematician. I appreciate that Wikipedia is about comprehensive, technical summaries of topics, and that there is always room for 'dumbed down' explanations in other parts of the internet, but having viewed the 'Talk' section on reaching non-mathematical readers, thought I'd pipe up with the question that many Wikipedia pages forget to elucidate: "Why". — Preceding unsigned comment added by Roamsdirac (talk • contribs) 8 May 2013‎ 07:57 (UTC)


 * Good point. I have edited the lead to take your remark into account. The way I have used seems better than your suggestion, because it is closer to the historical origin of the concept, at the true beginning of algebra. I have also added some clarifications in the terminology. D.Lazard (talk) 09:51, 8 May 2013 (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 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)
 * A non-polynomial algebraic expression perhaps, i.o.w. not a polynomial? See this edit - DVdm (talk) 11:06, 7 February 2013 (UTC)

It is a rational expression.Rick Norwood (talk) 17:38, 10 July 2013 (UTC)

Why complicate?
If Wikipedia is aimed at the general public, and not a highly specialized scientific treaty, why there are expressions like these: "... and also because its third term contains an exponent that is not a non-negative integer"... I mean, couldn't "is not a non-negative integer" be written as "is not a positive integer"? Are there any other options besides positive and negative? Gonello (talk) 12:31, 21 May 2013 (UTC)


 * Are there any other options besides positive and negative ? Yes - zero. A polynomial may contain one or more terms in which the exponent of all variables is zero - these are constant terms. So "non-negative integer" means "positive integer or zero". Gandalf61 (talk) 12:44, 21 May 2013 (UTC)

Definition of Polynomial
I'm wondering if the definitions for polynomial used on this page in the introduction (and elsewhere) might be erroneous.

Under this definition: "A polynomial is an expression constructed from variables (also called indeterminates) and constants (usually numbers, but not always), using only the operations of addition, subtraction, multiplication, and non-negative integer exponents (which are abbreviations for several multiplications by the same value)."

if you consider a polynomial with integer indeterminates (is this allowed?) x and y, wouldn't $$ x^y $$ be considered a polynomial? Bacawikiwawa (talk) 02:03, 7 June 2013 (UTC)


 * No. For a fixed non-negative integer value of y then $$ x^y $$ is a polynomial, so you could regard $$ x^y $$ as a family of polynomial functions indexed by y (as long as you restrict y to non-negative integer values). But considered as a function of two variables, $$ x^y $$ is not a polynomial. Gandalf61 (talk) 09:38, 7 June 2013 (UTC)
 * Moreover, one cannot talk of "integer indeterminate". An indeterminate is, by definition, a symbol without any value. Thus the correct formulation is: "If y denotes a nonnegative integer, then $$ x^y $$ is a polynomial in the single variable x, but, if y is an indeterminate, it is not a polynomial, not even an algebraic expression". D.Lazard (talk) 10:19, 7 June 2013 (UTC)

Somebody called quartic polinomial as "biquadratic". They are not synonims. Biquadratic is, for example, f(x) = x^4 + x^2 + 1. This one is not biquadratic: f(x)=x^3.

Use of Indeterminate
Is their any utility gained from distinguishing 'indeterminate' from 'variable'? After reading the article on indeterminates, I'm of the opinion that it's a useless, pedantic artifact---i.e., regarding footnote '5', maybe there's a reason no one bothers to draw a distinction between the two. — Preceding unsigned comment added by Jdc2179 (talk • contribs) 12:20, 7 June 2013 (UTC)
 * From a formal point of view there is no difference betwen "indeterminate" and "variable". Both are symbols that appear in a formula in place of some "value". Historically these values were numbers that were the argument of some functions. Therefore the name of "variable" for "which may vary". But in modern mathematics, the value of a variable may be any mathematical object and therefore the variable is preferably viewed as the "name" of the object which is represented (this is coherent with the meaning of "variable" in programming languages). For example, in the notation $$f(x)$$, x is a variable in the historical sense, and f is also a variable, as the name of a function (one would say a "functional variable"). Depending on the context, a variable may have several meanings and this is the reason for which some other words have been introduced to emphasize on a particular meaning, such that "unknown" for a variable in which an equation has to be solved, "constant" for a variable representing a value which is fixed in all the context where it appears, or "indeterminate" for a variable that represents nothing else as itself, as are the variables of a polynomial. D.Lazard (talk) 13:10, 7 June 2013 (UTC)

Even more fundamental, the set of all polynomials form a mathemetical structure called a Ring in which the "indeterminate" does not represent a variable, that is, in which x is not a symbol which holds the place of a number, but is rather just a symbol that follows certain rules such as x^n*n^m=x^(n+m). Calling something a veriable suggests it can take on many values. In f(x) = x^2, x is a variable. Calling something an unknown suggests that it can somehow become known. In x^2 = 36, x is an unknown. But if we never intend the x to be anything other than x, then it is better to call x an indeterminate. Rick Norwood (talk) 17:35, 10 July 2013 (UTC)

Section on polynomial interpolation/approximation
In case I don't get around to it, I wanted to suggest that it would be a very good idea to create a section on polynomial interpolation and other uses of polynomials to approximate more complex functions (Taylor's Theorem, etc.). -- Mesoderm (talk) 20:33, 10 August 2013 (UTC)

Question about cubic polynomials
can a cubic polynomial p(x) with integral coefficients satisfy the condition that p(a)=b,p(b)=c and p(c)=a ,where a, b and c are distinct integers? — Preceding unsigned comment added by 27.107.246.196 (talk) 05:27, 15 July 2013 (UTC)
 * There are a number of good sites on the web for asking mathematical questions, but Wikipedia is not one of them. Rick Norwood (talk) 12:25, 15 July 2013 (UTC)
 * Well, Wikipedia article talk pages are not one of them, but Reference desk/Mathematics is. -- Mesoderm (talk) 03:44, 11 August 2013 (UTC)

Recent edits
The recent edits may have made the article more accessible to lay readers, but they have also confused the definition of a polynomial so that the differences between a polynomial and a more general algebraic expression are no longer clear. I will not revert the edits, but I think the lead has to be reworked. The goal would be to improve the article accessibility while keeping any relevant details in the old version. Isheden (talk) 09:25, 23 October 2013 (UTC)
 * Also these edits have removed from the lead the fact that a polynomial may have several variables. Already the old version was not very clear with that, but this is worst with the new. Wikipedia is not written only for kids and college students. It must also be written for engineers and non-mathematician researchers that may encounter multivariate polynomials. Presently, somebody that search for "polynomial in several variables" or "multivariate polynomial" will only find, after reading carefully the section "Definition" up to the end that they exist, but they will not find any definition of the notion nor the location of such a definition nor any description of their properties. This has to be corrected. D.Lazard (talk) 09:56, 23 October 2013 (UTC)
 * You say that it MUST be written for engineers and non-mathematician researchers, but aren't these specialist audiences, and is there any evidence that this group of readers are in need of advanced mathematical treatise? The issue that this article had was that it doesn't address the needs of the specialist, and it is indecipherable by everyone else. John lilburne (talk) 10:25, 23 October 2013 (UTC)

I'd say polynomials in several variables need to be defined and described in the article, but that does not mean that they must be included in the definition in the first sentence of the lead. Presently, the lead contains $$x^2-2xy+y^2$$ as example of a polynomial, which is unfortunate. In my view it is more important to distinguish (univariate) polynomials from more general algebraic expressions such as $$\sqrt{x^2 + 1}$$ and to give a few examples of polynomials of different degrees. Furthermore, the present lead might give the impression that $$(x+1)^2$$ can be considered as a polynomial. I think the MathWorld entry on polynomials has a better definition: "A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients." Isheden (talk) 11:39, 23 October 2013 (UTC)
 * If people are given the definition of what a polynomial is and an example, it is self evident what a polynomial is not. Listing what is not a polynomial is superfluous. I could understand if we were writing an excercise book for 12 year olds, but we aren't. IRWolfie- (talk) 14:07, 24 October 2013 (UTC)
 * In this case an experienced math teacher should be able to tell whether or not counterexamples help improve the understanding of the distinction between polynomial and algebraic expression. Suffice it to say that in the version I was referring to, the definition in the first sentence would include the expression $$\sqrt{x^2 + 1}$$ and the second sentence would exclude the expression $$(x+1)^2$$. Isheden (talk) 14:46, 24 October 2013 (UTC)


 * I'm guessing these edits have something to do with the critique of the article at http://wikipediocracy.com/2013/10/20/elementary-mathematics-on-wikipedia-2/. Someone not using his real name (talk) 11:53, 23 October 2013 (UTC)
 * Correct. See . Isheden (talk) 11:59, 23 October 2013 (UTC)
 * Yes. IRWolfie- (talk) 09:55, 24 October 2013 (UTC)

I am all for making Wikipedia articles accessable, but not at the expense of accuracy. The lead must give correct definitions. Rick Norwood (talk) 12:07, 23 October 2013 (UTC)


 * Someone may be interested in also attracting a mathematician to the topic by informing wikiproject maths. I will do so myself later, IRWolfie- (talk) 10:59, 24 October 2013 (UTC)


 * There are several regular editors of mathematics articles in this discussion already, but I have posted a note at Wikipedia talk:WikiProject Mathematics anyway. Gandalf61 (talk) 12:03, 24 October 2013 (UTC)


 * Rick Norwood: the purpose of the lead is to introduce and summarize the article. As in the case of research papers, this sometimes should involve precise definitions and sometimes not.  The blanket statement that the lead must give correct definitions is just wrong. --JBL (talk) 14:03, 24 October 2013 (UTC)


 * Err what? Are you saying it's ok if the lead contains incorrect definitions? IRWolfie- (talk) 14:04, 24 October 2013 (UTC)


 * I'm saying that it's okay if the lead of some math article does not contain a fully precise, formal definition. Whether that's a "yes" or a "no" may depend on your personal perspective.  --JBL (talk) 14:11, 24 October 2013 (UTC)
 * I think having both the formal and informal usage is the best approach, then both types of readers get what they want out of the lead. i.e like how the Encyclopaedia Britannica have done it. IRWolfie- (talk) 10:54, 25 October 2013 (UTC)

Lead sentence
I have a problem with the current first sentence of the article and actually with the lead in general. The first sentence basically refers to any algebraic expression that only contains integer exponents. Thus, for example $$(x+1)^2 + (x-1)^2$$ would be referred to as a polynomial. I'm not sure if this would normally be considered a polynomial; in that case at least a reliable source should be provided. Further down the lead refers to such an expression as a polynomial expression, but it is not even clear (at least to a high-school student) if this definition is consistent with the one in the article polynomial expression "any meaningful expression constructed from copies of those entities together with constants, using the operations of addition and multiplication". As mentioned in the previous section, I think the MathWorld definition "A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients" is more accurate. Isheden (talk) 12:44, 24 October 2013 (UTC)


 * The question you ask is whether a polynomial is a form or an algebraic expression. You suggest that we can have two equal expressions, and one is polynomial in form, the other not.  This is not the way the term is actually used.  A polynomial may be in expanded form, in which case it is a sum of (positive integer) powers with coefficients, but it may be in factored form or, as in the example you gave, partially factored form.  But the set of polynomials forms a ring and in order for multiplication in the ring to be well-defined two equal expressions must both be polynomials if one is a polynomial.  I don't there is one definition in high school and a different definition in college, though that problem does arise and is currently being discussed over in fraction.Rick Norwood (talk) 12:54, 24 October 2013 (UTC)
 * In that case I don't see why the lead should distinguish polynomial expressions from polynomials. Why not just give examples of polynomials in expanded, factored, and partially factored form? Presently only the factored form seems to be mentioned in the body of the article. Isheden (talk) 13:28, 24 October 2013 (UTC)


 * Is it an analogous problem to ask if 8-1 is an integer or if only the reduced expression 7 is called an integer?--LutzL (talk) 13:27, 24 October 2013 (UTC)

Some editor was overly fond of the expression "expression". I've taken out some of them. Rick Norwood (talk) 19:21, 24 October 2013 (UTC)


 * Strictly formally, a polynomial in x is an element of the ring extension R[x]. All expressions and sums of monomials are then "only" different representations of a polynomial, where the sum of monomials is usually considered as normalized or standard. However, $$((x+1)^{10}+1)^{10}+1$$ represented as sum of monomials loses some of its expressiveness.--LutzL (talk) 20:17, 24 October 2013 (UTC)
 * The present sate of the lead looks rather nice. Unfortunately it is mathematically wrong: a polynomial is not an expression. In fact, $$x^2-1$$ and $$(x+1)(x-1)$$ are not equal as expressions, but are equal as polynomials. Thus a polynomial may not be an expression without making mathematics contradictory. I see two solutions to solve this issue. Either, adding "roughly speaking" in the first sentence, and creating a new section explaining this and providing a correct definition. Or replacing polynomial by polynomial expression in the first sentence and adding something like:
 * "A polynomial is the class of all the polynomial expressions that may be rewritten to a given polynomial expression by using commutativity, associativity and distributivity. Any element of the class may be used to designate the polynomial."
 * I am not fully satisfied by either solution (otherwise I would be proud and insert it). Maybe someone would find a better formulation. In any case, we can not keep a formulation which is mathematically incorrect in this important article. D.Lazard (talk) 10:30, 25 October 2013 (UTC)


 * This would lead back to the question if $$8-1=7$$, both sides are not equal as expressions but equal as integers. Usually and intuitively, equality is understood as semantic equality, and not syntactic equality. If not, then one would also have to deal with the situation that $$x^2-1$$ and $$-1+x^2$$ or just $$x^2+(-1)$$ may be considered as syntactically different expressions.--LutzL (talk) 11:24, 25 October 2013 (UTC)
 * The point D. Lazard raises is actually what I was trying to hint at. This technicality is probably why polynomials are often described as a certain expression, namely the expanded form with the terms ordered by power, although for example a partially factored form represents the same polynomial. Isheden (talk) 14:17, 25 October 2013 (UTC)

I think most of the problems with the lead have been resolved now. I'm still a little bothered by the statement that n is the degree of the polynomial, but since earlier it is stated that "the" polynomial is a polynomial in one variable, I guess talking about the sum of the exponents on variables in one term would needlessly confuse the issue. Rick Norwood (talk) 12:08, 25 October 2013 (UTC)
 * Now that most problems have been resolved I think it's time to expand the lead a bit to summarize some of the most important aspects of the subject. It could be mentioned that sums and products of polynomials are also polynomials, what the fundamental theorem of algebra states about the numbers of roots, that a partially factored form may be irreducible over the reals or rationals, that there are formulas for solving polynomial equations up to a certain degree... Isheden (talk) 14:29, 25 October 2013 (UTC)

Etymology
Once the origin of the word was taken verbatim from etymonline.com without attribution. This was changed to the current state, which is incomplete in only explaining the first part. The common understanding on the internet can be found in e.g. this survey on mathforum.org. The only source given for the origin is (Cajori 1919, page 139) from "Earliest known uses". This (correctly) refers to Florian Cajori (1919): "A history of Mathematics" archive.org. A later book, F. Cajori (1928/29): "History of mathematical notations", covers nearly the same topics and provides lots of reproductions of original manuscripts and prints. On the origin of "polynomial", especially in its current use, these books are still inconclusive, but an approximate history can be established:


 * Fibonacci in translating arabic sources used or even invented "binom*" as in "root of a binomial" as a technical terminus. Until modern times, "binom*" was used for the aggregation, mostly as sum, of two irreducible terms, like $$2+\sqrt3$$, that are the argument for a root or an integer power. This use and origin strongly indicates that "nom" is from latin "nomen=name" and not greek "nomos=law". Somewhere along the way, "trinom*" was introduced for sums of three terms. Roots of sums of more terms were labeled as "univers*", in german lands also "collect*" and later "legato". For instance, "R.bin.2.p.R3" could stand for $$\sqrt{2+\sqrt3}$$.


 * The use of these terms became obsolete, but never completely vanished, by the introduction of other grouping notations, such as "√ʒ.2+ √3.", overlines and much later, parentheses.


 * Around 1585 (the reproduced print in Cajori (1928/29) is from 1634), Stevins in a french text used "multinom*" as generalization of "binom*" and "trinom*" (the use of "monom* or mononom*" would have been absurd in this context).


 * Cajori (1919) claims, without providing context, that Viete in a text from 1591, aside from inventing "coefficient", also used the term "polynom". Cajori (1928/29) makes no mention of that. If this reference is correct, it can be guessed that Viete knew about the text of Stevin and altered "multinom*" to "polynom*" for essentialy the same usage, roots and powers of aggregated terms. Since Viete introduced the use of parameters, the terms in a "polynom*" were no longer restricted to only numbers and roots of numbers, also expressions containing these parameters could occur. However, there is no guarantee that these general algebraic expressions were always multivariate polynomials in the parameters.


 * The first documented use of "polynomial" (and this in the modern sense) in Cajori (1928/29) is from 1808. Before that, polynomial equations were always "equations of order (2,3,4)" or "cubic equations" etc., polynomial functions were "numeric functions" as used by Galois or in latin texts "integral rational algebraic function" as in Gauss (1799) "...functionem algebraicam rationalem integram unius variabilis..."

It would be nice if someone could access the source of the Viete text, the manuscript of 1591 or the print of the collected (and notationally reformulated) works by Schooten 50 years later, to verify if and how "polynomial" was used. Further insight into the first use in the present meaning would also be welcome.--LutzL (talk) 22:38, 26 October 2013 (UTC)

Mentioned in blog
This article has been mentioned in Wikipediocracy: RockMagnetist (talk) 22:18, 8 November 2013 (UTC)


 * OK. Are you after help going through gutting the article of excess crap? If it's interesting excess crap, it can be relegated to between . LudicrousTripe (talk) 22:44, 8 November 2013 (UTC)


 * I might do a bit, if I can find the time. You have already made some improvements since that blog was posted. RockMagnetist (talk) 22:55, 8 November 2013 (UTC)

Polynomial vs. polynomial function
It might be useful to distinguish between polynomials and polynomial functions. The example of x^p - x and 0 over the field of integers mod p, a prime, shows that equality for polynomials and functions is different. A reference, should one be required, is Birkhoff & Mac Lane, 3rd ed., Ch III, section 2. 86.132.223.116 (talk) 23:45, 9 December 2013 (UTC)

Definition
There is more than one definition of polynomial. The one given in the article is inadequate. For one indeterminate consider, e.g.:

i) Integral domain E containing x, the indeterminate, and a subdomain containing the coefficients..., or

ii) A polynomial is a sequence of elements from an integral domain (or other suitable structure) of which only a finite number are non-zero... 86.132.223.116 (talk) 00:07, 10 December 2013 (UTC)

Language problems
I just went to this article to refresh my knowledge. I immediately noticed several instances of improper English usage, that appear to have been written by a non-native speaker. I am reluctant to perform much editing in an important article when I have only a dilettante's knowledge of the subject.


 * Examples:

The word indeterminate is used often; the more common (and only correct) synonym is variable.

Under 'Notation', the parenthetical sentence contains it is a common convention of using upper case letters.... This should be something like it is a common convention to use upper case letters.... This verb form confusion is common when writing English translated from other languages.

Again under 'Notation', in the last paragraph; This equality allow to simplify wording in some cases.... The verb 'allow' is in the wrong form, it should be 'allows' - but the entire sentence is subtly wrong.

Same paragraph, much easy to read should be much easier to read.


 * I hope I've stated my thesis without being too wordy.

Trelligan (talk) 04:57, 16 December 2013 (UTC)


 * ✅. Good find. But you could have done this yourself :-) - DVdm (talk) 07:46, 16 December 2013 (UTC)


 * I agree with most corrections, but the systematic replacement of "indeterminate" by variable deserve a more careful discussion. The assertion
 * "The word indeterminate is used often; the more common (and only correct) synonym is variable."
 * is not correct in mathematics. I agree that in correct English "indeterminate" is not a noun and that its usage as a substantive is incorrect. However, in current mathematics "indeterminate" is a noun that may be sometimes, but not always, replaced by "variable". Formally speaking (that is from the point of view of mathematical logic), the X that appears in a polynomial is not a variable, but a constant of the theory. However, as many people (including all mathematicians until the end of 19th century) do not clearly distinguish a polynomial from its associated polynomial function, the X or x appearing in a polynomial and its associated function was commonly called "variable". But, nowadays, "indeterminate" is the only mathematically correct word to denotes the X of a polynomial. In many cases, the abuse of language of using "variable" instead of indeterminate is not confusing, but it is in some cases. In particular, DVdm's edit has changed
 * "it is a common convention to use upper case letters for the indeterminates and the corresponding lower case letters for the variables (arguments) of the associated function"
 * into
 * "it is a common convention to use upper case letters for the variables and the corresponding lower case letters for the variables (arguments) of the associated function"
 * which is correct English but does not mean anything.
 * I note that after DVdm's edit, the article keeps a correct definition of the noun "indeterminate" ("consisting of variables, called indeterminates" in the lead, and "x is a symbol which is called indeterminate or, for historical reasons, variable", in the section "Definition"; here articles are lacking before "indeterminate" and "variable"). As WP must avoid, as far as possible, abuses of language, my opinion is that "indeterminate" must be used systematically. However, as this usage is not common at elementary level, it should be clarified in a more visible way that "variable" is commonly used instead of "indeterminate", but that this terminology may be confusing. I'll thus revert the replacement of "indeterminate" by "variable" and try to elaborate this clarification. D.Lazard (talk) 10:32, 16 December 2013 (UTC)

Dictionaries often do not include technical terms, which are nevertheless good English. Lazard's explanation is correct. Rick Norwood (talk) 19:48, 16 December 2013 (UTC)


 * Sure. Whatever makes us all sufficiently happy, is fine with me. I think the article got better thanks to Trelligan's remarks, even if "indeterminate" is a term I never came across in my—non-English—math education. - DVdm (talk) 19:58, 16 December 2013 (UTC)

Calculating and Computing
Request for comment and help editing/ resoring correctly. Do not edit/remove sections until we hear from other editors during this talk. Statements by Lazard removing this section for "already covered" are incorrect. There was no other mention of Stone–Weierstrass theorem or its applications in the entire article. Section can use help for sure, but outright deletion/ edit war with no comment or discussion is negaquitte!Pdecalculus (talk) 17:42, 26 December 2013 (UTC)


 * Stone-Weierstrass theorem appears in section "Calculus". The sentence "Polynomials are the only everywhere differentiable functions that can be directly calculated with electronics" is an unsourced controversial assertion which is WP:OR, because "directly calculated with electronics" is a notion which is not defined elsewhere. The remainder of this section is an original synthesis, about how polynomials are used in computer programs. The fact that it is an original synthesis appears clearly by the fact that, instead of referring to a source, the reader is invited to "directly research and examine open source equivalents" or "examine the code", or even to believe what is said about of the "trade secrets of software companies". This reference to the trade secrets of software companies suffices to show that the author of this section does not know much on this subject: the state of the art applications of polynomials to computing are all published in academic journals, even the few ones that are developed in software companies and the few ones that have led to patents (yes, there are patented polynomials). It follows from these facts and Wikipedia policy that this section must be removed as blatant original research. D.Lazard (talk) 22:20, 26 December 2013 (UTC)


 * Agreed. Original research can't get more obvious than "Polynomials are the only everywhere differentiable functions that can be directly calculated with electronics, because, by definition, they only involve addition, multiplication, subtraction and exponentiation. Because of this, polynomial approximation is used in many computing platforms, from the simplest calculators to Excel, Maple, Matlab, Mathematica, Fortran and Haskell, for example, to indirectly calculate other values, such as trig functions or logs. In commercial applications these functions are often proprietary, but can be directly researched and examined in open source equivalents such as GNU Octave." - DVdm (talk) 10:34, 27 December 2013 (UTC)


 * First, you are not supposed to remove sections during active talks. Second, Lazard cracks me up with his OR concept, the quote he used as an example is nearly an exact reference from Dr. Jim Stein, How Math Explains, the World, p. 82. "Blatant" original research is even funnier given that Stein gives several bib/journal references for the statements. The sad thing is that I get continual solicitations from Wikimedia to act as a math expert for them. Then, I have to deal with folks who come up with logic like this! I was not finished adding references when the whole section was deleted, again, against Wiki policy during active talks. I want someone to weigh in who understands the importance of polys in computing before less expert editors make these decisions.


 * I'd appreciate additional comments from folks who understand numerical methods and polynomials in their role in computing. These are NOT the province of "calculus" in this frame, but are much more precisely represented in numerical methods, CAS, etc. In fact, I'd suggest a whole ARTICLE on Polynomial computing, yet a couple folks here can't even see a section!! Before agreeing with Lazard outright, as one editor did here, mistaking a direct, referenced quote for original research, let's hear from some folks working, as I am, IN polynomial computing. Lazard's personal attacks and "the editor doesn't know much" might make him feel his Wheaties ego, but I've taught graduate numerical methods for 30 years. "yes there are patented polynomials (SIC)" stated by Lazard with amazement and ego... What hubris, and how condescending! I OWN Payroy.com and WE patent hundreds of math formulas each year! Thanks, and if anyone can ignore the non constructive attitude about OR, and the mistakes I've apparently made, I'd be happy to contribute to a whole article on polynomial computing, as there are numerous texts written, and being written on the topic, including one I'm editing right now. Pdecalculus (talk) 16:20, 27 December 2013 (UTC)
 * I agree that this article deserves to have a section on "polynomials in computing". However your text appears to be a tentative to summarize the mathematical results that are used in computers and that involve polynomials, directly or indirectly. IMO, the very high number of such mathematical results makes such a summary almost impossible, for a weak encyclopedic interest. On the other hand, as polynomial are widely used in computers, it is important to describes how they are represented in computers (dense and sparse representation, and their variations for multivariate polynomials), evaluated (Horner scheme) and manipulated (fast and standard polynomial arithmetic, polynomial root finding, etc.). Presently, Wikipedia is very poor on these questions, although there are fundamental. In fact the implementation of any mathematical result involving polynomials depends first of the choice of some solution for these fundamental questions on polynomials. The choices that are usual for implementing a particular mathematical result must be described first in the article about this result, even if a link to this article may deserve to appear in the description of the general polynomial method. D.Lazard (talk) 17:50, 27 December 2013 (UTC)


 * Sorry that you're experiencing some turbulence in your attempts to contribute to Wikipedia. I can't judge the technical merits of your contribution, but it does seem to me that there is nothing wrong with it that can't be fixed by a few citations. I never add content to an article without the citations already embedded, and I can assure you it makes life easier. You might want to consider composing it in your sandbox, complete with citations, before adding it to the article.


 * Your "nearly exact reference" raises a warning flag. Beware of copying text or close paraphrasing it, or you might run foul of Wikipedia's Plagiarism policy. Also, the style of a book like How Math Explains the World is not encyclopedic, and that is probably part of the reason DVdm and D.Lazard perceived it as original research. RockMagnetist (talk) 17:54, 27 December 2013 (UTC)


 * Yep. - DVdm (talk) 18:02, 27 December 2013 (UTC)


 * Thanks, didn't know we couldn't use quotes from books. Subject change: for anyone interested in polys I'm reading this interesting new algo/software title right now which you might enjoy: "Numerically Solving Polynomial Systems with Bertini.." ..."approaches numerical algebraic geometry from a user's point of view with numerous examples of how Bertini is applicable to polynomial systems. It treats the fundamental task of solving a given polynomial system and describes the latest advances in the field, including algorithms for intersecting and projecting algebraic sets, methods for treating singular sets, the nascent field of real numerical algebraic geometry, and applications to large polynomial systems arising from differential equations." BTW, THANKS FOR THE ADVICE ON INCLUDING THE CITES right up front. I've done this on whole articles but you have the under construction option there which allows other editors to cut a little slack until you get it right. I'm going to retry this section as an entire article, then if it survives, use it as just a section here. I envision it being specifically on polynomial computing, drawing from the related fields of numerical methods, etc. While the present article mentions computational complexity, as most folks here know there is a great difference between 1. NP type uses of polynomial methods in CC vs. 2. Solving polynomial systems WITH numerical methods vs. 3. USING polynomials ala Stone-Weierstrass theorem and many other algos in CAS, ALUs, etc. I work with TI and HP all the time, and there are some awesome polys in the newer logic circuits that estimate the trig of circuit cross sections and voltages as numerical signals, very cool stuff. BTW, check out the new HP Prime calculator while on the topic, although it is weak in RPN, polys are handled very sweetly and the gui is like this younger generation's smart phones! — Preceding unsigned comment added by Pdecalculus (talk • contribs) 18:58, 27 December 2013 (UTC)
 * "Bertini" is a software that deserve to have an article in WP. However, polynomial system solving is a wide area that does not reduces to this software. You may enjoy in reading System of polynomial equations, which is rather complete for the case of the systems which have a finite number of complex solutions. D.Lazard (talk) 19:32, 27 December 2013 (UTC)


 * Great idea, not many folks know about it. Barbeau is mentioned in the article and I enjoy the breadth of problems in his book. I'll check out the systems article too, thanks. — Preceding unsigned comment added by Pdecalculus (talk • contribs) 23:56, 27 December 2013 (UTC)

Examples? Tutorial? Explanation to general audience?
So many of these science, tech, and math (especially math) articles are written exclusively to an audience that already knows the material – people who don't need the article in the first place. In order to be truly useful wikipedia must be more than an archive of compressed insider information. I am more and more of the opinion that this problem is (at least partially) rooted in the actual editing environment (the editing interface) which feels custom built to exclusively filter for a particular class of computer nerd, a class of people who are notoriously uninterested in teaching or of expending any effort to make technical information cognitively available to non-nerds. This filter excludes the very category of potential contributors that could transform wikipedia from an archive of coded insider information - and into a true information transference medium. The entreaty, "Did you know that you can edit this page?" isn't honest. Only those with HTML editing experience will be comfortable contributing within the wikipedia page editing environment. This is either a gross oversight (highly unlikely), or a persistently and aggressively arrogant form of direct and purposeful filtering of potential contributors. Feels for all the world like social engineering gone terribly wrong (is there some other way for social engineering to go?). Change the interface to a simple and direct WYSIWYG content editor and watch as wikipedia content becomes friendlier and more instructive.

Note: I realize that much work has been done to make the wikipedia editing and contribution environment more "humane", but the legacy of nerd-facing tools and attitude presents a considerable wall to overcome. Wikipedia fights an uphill battle to attract and keep the non-nerd contributor. The overall look and feel of the wikipedia interface seems very much more an attribute of the underlying page description protocol and very much less an attribute of the purpose - that of making it intuitive and slippery smooth for ANYONE to contribute and edit content.

Randall Lee Reetz — Preceding unsigned comment added by Randall Lee Reetz (talk • contribs) 17:57, 20 May 2012 (UTC)

Randall,

While I think there are some valid points in your dissertation, it is not part of a discussion about Polynomial Expressions. You would probably get a larger audience and some support if you considered a different location. Les Hayden (talk) 13:55, 3 June 2012 (UTC)

D. Lazard Operators Questions
Dr. Lazard: In working on a polynomial computing article I was shocked to find there also is no article or section on Polynomial operators! As I'm sure you know, many researchers and students, in structuring or working on linear equations and nonlinear systems, fail to notice the underlying polynomial nature of the problems. Multivariate polynomial operators are very important in polynomial computing, special functions, numerical methods and other algorithms for CAS, but I think they might be less known when seen only as a footnote to tough nonlinear systems or even linear equations, which of course are actually a subset of PO's. Do you think an article or at least a section is warranted? Recent work also is applying sytems of PO's to Latin squares for new approaches to research design methodology (as a statistical generality / approach). Pdecalculus (talk) 16:41, 29 December 2013 (UTC)


 * Thanks to db for comment on being careful of techniques. I've been zinged before on articles for giving technique instead of theory because there is a wiki caveat against "how to's" I guess. Because of this I'm shy about applications, but have combed a bunch of them on the site looking for (not totally how to) connections between polys, computing and linear algebra, with a view toward perhaps compiling a list of polynomial operators if not an article. These include but are not limited to related articles on linear algebra, polynomial algorithm generation, functional methods and shift operators, signal processing, of course numerical LA, Bezoutian and Hankel forms (eg. what wiki would call polynomial stability testing ala Bézout matrix or Hankel matrix-- although even the ortho poly section there is incomplete), control theory, etc. On the current wiki, the relevant operators are scattered thoughout the site, some with individual articles, others buried "as" an application. I don't know the site rules enough to know if a comprehensive article explaining them, or a list tying them to polynomial computing and algos is warranted, and I don't want to spend a week writing it if it's just going to be removed. I don't mind it being heavily edited, but removal just kills the idea under the rubric it's covered elsewhere, which is true, but you need a trail of bread crumbs to find it!Pdecalculus (talk) 17:25, 29 December 2013 (UTC)


 * Forum update db: I'm cracking up at your idea that a list obviates the covered elsewhere argument because, by nature, a list IS what's "covered elsewhere." If other editors agree, maybe I'll start there! And thanks for the encouragement. — Preceding unsigned comment added by Pdecalculus (talk • contribs) 22:19, 29 December 2013 (UTC)


 * Calc: thanks! I've moved my operator efforts to Operator (mathematics) which had ZERO references or cites, and no mention of polynomials. It's pretty much an orphan.Pdecalculus (talk) 21:41, 3 January 2014 (UTC)
 * An orphan is an article without incoming links. With more than 50 incoming links (their list may be obtained with the button "What links here" at the left of the page), Operator (mathematics) is far to be an orphan. D.Lazard (talk) 16:54, 6 January 2014 (UTC)

Section Polynomial equations
At present, this section has two subsections. Allegedly, the division between the subsections is between introductory material and material on solving polynomial equations. In fact, both sections contain definitions of polynomial equations and both sections contain material on solutions of polynomial equations, but the notation, style, level, etc. is completely different between the two sections. (In particular, the first section is much more approachable and does not introduce unnecessary subtleties like the difference between a polynomial and its associated function.) I tentatively propose to rewrite this section to remove redundancy; I will probably aim for a level closer to the level of the first, smaller, subsection. What do others think? --JBL (talk) 23:25, 12 June 2014 (UTC)

Reciprocal equations
This is a term mentioned in some good treatments of polynomials yet it doesn't seem to be mentioned on wikipedia. Would it be appropriate to mention it on this page? If so, how should it be incorporated?

SewerCat (talk) 18:51, 21 September 2014 (UTC)
 * See Reciprocal polynomial. D.Lazard (talk) 19:30, 21 September 2014 (UTC)

Merger proposal
I propose that Polynomial expression be merged into Polynomial. This is a bad way to distinguish the two terms "polynomial expression" and "polynomial"; even if there were a better title for the first article, it's unlikely to be expanded. (It was created in 2010 by one user and has received no substantial edits since that day.) I think that its content could go well at the end of the Definition section here; we wouldn't need the introductory context, just a simple statement and the examples. (On a related note, the entire Definition section here needs refactoring, but that's a bigger job, and the refactoring will go better if the merging is done first.) —Toby Bartels (talk) 16:07, 13 April 2014 (UTC)


 * Since I was that user, I just like to let know that I don't oppose to the proposed merge. A main motivation for creating the polynomial expression article was to have a destination for links that would not just (and confusingly) point to polynomial; however this concern might be taken care of by a specific redirect (if a target is available). I think the term "polynomial expression" is fairly well understood by mathematicians in the field, any often freely used without defining it; as it is not something one writes books about, so it does not surprise me that finding sources is hard/impossible. Sometimes it is used without the "expression", although this usage contradicts the definition of polynomial (see for instance matrix polynomial). Good luck with trying to find the right angle/tone for discussing this in the context of the Polynomial article (which is more aimed at a broad public, I think).Marc van Leeuwen (talk) 16:57, 13 April 2014 (UTC)


 * Support: I support the merge. However, care should be taken that polynomial expression has two slightly different meanings. The article to be merged here call polynomial expression the result of substituting the indeterminate(s) of a polynomial by some mathematical object (called here "entity"). As far as I know, in most mathematical texts, "polynomial expression" is not used for this purpose, but the authors use instead something like "polynomial in cos(x)", for example. In other contexts, "polynomial expression" is used for expressions that may eventually be simplified into a polynomial, while "polynomial" is reserved for the expanded form of the polynomial. D.Lazard (talk) 17:38, 13 April 2014 (UTC)


 * I share D.Lazard's concern about the two different senses of "polynomial expression". (Actually, I'm not sure that I'm familiar with either sense, Marc's or Lazard's, as a term to be used specifically in preference to "polynomial", although in both cases I can see that it's being used for something slightly different than a polynomial exactly.)  This means that redirecting polynomial expression to a place within the polynomial article wouldn't work either.  Perhaps there could be some very brief mention of Marc's sense of "polynomial expression" in the lede.  —Toby Bartels (talk) 03:04, 17 April 2014 (UTC)


 * OK, here's another idea. Maybe we have a little section in this article discussing the term "polynomial expression" in all of its senses.  This would be in addition to merged text handling things like polynomials in cos&thinsp;x and matrix polynomials (not to mention the current material handling expressions that reduce to polynomials upon algebraic simplification).  But then we would need citations.  —Toby Bartels (talk) 03:09, 17 April 2014 (UTC)


 * I support this proposal. --JBL (talk) 23:27, 12 June 2014 (UTC)


 * I'm prepared to sit on board with this proposal as well, although I'm still a little skeptical about referring to polynomial-like expressions as "polynomial expressions" (but then that's just me). --JB Adder | Talk 22:15, 13 September 2014 (UTC)


 * Support: As far as I remember, I've always seen polynomials and polynomial expressions in the same section in a majority of my mathematics text books. As far as Polynomial expression is considered as a standalone article, it's still a stub and viewing from an encyclopaedic point of view, I do not see anything more meaningful that could be added, other than stuff for highly targeted audiences. However, as D.Lazard mentioned above, care must be taken to clearly mention the contexts of usage of the term "polynomial expression"  Amitosh.swain (talk) 05:57, 1 November 2014 (UTC)

Polynomial Calculator External Link
Hello Sirs,

Thanks for the explanation. Those software are kind of charity provided for free usage of public. I believed could be helpful promoting them in a convenient place frequented by relevant pupils and scholars. I couldn't find any other way.

Therefore I insist putting it here improves useability of wikipwdia entry.

Regards 86.31.47.92 (talk) 11:33, 1 November 2014 (UTC)
 * The explanation referred to in the preceding post is here, and concern the inclusion of this external link by this and this edit. I have reverted these edits, as, IMO, they do not satisfy the requirements for an external link. Other opinions are welcome. D.Lazard (talk) 14:29, 1 November 2014 (UTC)
 * I agree with D. Lazard. --JBL (talk) 14:56, 1 November 2014 (UTC)
 * Sirs, it is similar to Derek's Virtual Slide Rule Gallery link that I found on Wikipedia Slide Rule page. Were not that link on Wikipedia I never could know such a thing should exist by software. That slide rule is in java script codes. The Dysprosium Polynomial Calculator Software is in Java.86.31.47.92 (talk) 18:34, 18 November 2014 (UTC)

Number of coefficients in multivariable polynomials
I think this article would benefit from mentioning the maximum number of terms a multivariable polynomial may have. I've seen a proof using the stars and bars method from combinatorics that shows it is something like: $$\binom{n+d}{d}$$ for a polynomial of degree $$d$$ with $$n$$ variables. I do not have any sources, but I assume someone can dig this up. — Preceding unsigned comment added by 65.129.216.147 (talk) 02:50, 10 February 2015 (UTC)
 * You are right: $$\binom{n+d}{d}=\binom{n+d}{n}$$ is the number of monomials of degree at most d in n variables. This is said in Monomial. This would be worth to mention this here, as well as in Polynomial ring and in Monomial basis. D.Lazard (talk) 09:16, 10 February 2015 (UTC)

Proposed sections
I came to this article to refresh myself with basic operations working with polynomials. I found the addition section informative but then could not find information on the subtraction, multiplication or the factoring of polynomials. Also, the extensive descriptions were a little too technical for my, i.e., not really understand for me the layperson. I would like to add this sections if this is acceptable to the contributors/editors to this article.


 *  Bfpage &#124;leave a message 12:36, 11 August 2015 (UTC)

Etymology
Greek word for polynomial is πολυώνυμον (polyonymon) that means "something with many names" from πολύ = many and όνομα = name. Obviously polynomial is derived from πολυώνυμον. Georges Theodosiou, 86.204.67.10 (talk) 09:40, 2 November 2015 (UTC)
 * This could be plausible. Unfortunately this is wrong, as, historically, the word has been introduced in Latin before being translated in Greek. In fact, ancient Greeks did not have the concept of polynomial, and the word dates from the 16th century (to be checked), when the language of mathematics were Latin. D.Lazard (talk) 11:20, 2 November 2015 (UTC)


 * Anyway polynomial is derived from ancient and modern greek words πολύς (polys) = many, much, and ὄνομα (onoma) = name, in french nom. Georges Theodosiou email: chretienorthodox1@gmail.com 92.161.28.103 (talk) 12:55, 18 November 2015 (UTC)
 * You are certainly right for "poly", but completely wrong for "nomial". The word "polynomial" has been derived from Latin "binomius" during 17th century (see ). Do you pretend that the "bi" of binomial has a Greek origin? Moreover, as the Latin and the Greek words for "name" are similar and have the same Indo-European root, one may not pretend that the Latin word is derived from the Greek without a careful historical study. Do you have a reliable source for that? If not, your assertions are only your personal belief and are not reliable. D.Lazard (talk) 15:06, 18 November 2015 (UTC)

Definition

 * Current opening line of section Polynomial: A polynomial is an expression that can be built from constants and symbols called indeterminates or variables...
 * My response to this edit, resullting in "A real polynomial is one with all real constants and where the variable is restricted to the real numbers" with a source saying "When the coefficients are real numbers and the variable is restricted to the reals, the polynomial may be addressed as a real polynomial."


 * According to this must be rubbish so, "fixing terminology (a polynomial is not a function)" resulting in unsourced "A real polynomial is a polynomial with real coefficients", leaving a source in place that says differently.
 * Please read what it says: "it didn't or doesn't say that a polynomial is a function. See cited source. And see first line of this section, which also mentions the "variable"".
 * According to D.Lazard: "By definition, the variable or indeterminate of a polynomial has no value. Thus the variable of a polynomial may not be restricted to anything"
 * So, I have removed the source:, , leaving the phrase A real polynomial is a polynomial with real coefficients.
 * Comments welcome. - DVdm (talk) 11:24, 25 December 2016 (UTC)

Remarks on solving equations section
Might say something about trigonometric solution of cubic equation.

ETBell Men of Mathematics chapter on Hermite says that there is an analogous solution of the quintic in terms of elliptic modular functions, and that Poincare has found a general solution for higher degree equations in terms of functions of N complex variables having 2N periods. Is Bell correct? — Preceding unsigned comment added by 209.159.232.121 (talk) 16:42, 5 December 2017 (UTC)

Details about solving cubic equations are in Cubic equation, and trigonometric solutions are in Cubic equation. For quintic equation, look at Quintic equation, specifically Quintic equation. See also Hilbert's thirteenth problem. D.Lazard (talk) 18:21, 5 December 2017 (UTC)

Polynomial functions
is, since a long time, a redirect to the section "Polynomial functions" of this article. Recently has transformed the redirect into an article that contains nothing else that this section. This is a clear WP:CONTENTFORK, and I have reverted it. restored his edits, and has been reverted by another editor.

I hope that will not continue this starting edit war. In any case, I recall that such a modification of the structure of Wikipedia requires a WP:consensus on this talk page. D.Lazard (talk) 09:39, 8 April 2018 (UTC)
 * I will not, don't worry. Concerning the discussion, I want to point out that polynomials and polynomial functions are NOT the same thing. For instance, in Z/2Z, the polynomial X(X-1) is not zero but the associated polynomial function is identically zero... It seems to me that this must be at least explain in the main article polynomial. Valvino (talk) 09:55, 8 April 2018 (UTC)
 * A counterargument could be WP:Bold. It could be worth having a separate Polynomial functions article if there is more material to add which would make the Polynomial article too long. Jonpatterns (talk) 09:58, 8 April 2018 (UTC)

Polynomial functions, the II.
Given the definition of polynomial functions in this article via evaluating a polynomial, it takes me by surprise that my edit was reverted by giving as reason not "it takes the values of the polynomial function", they are the same function!

The intention of my edit was to give both a comprehensible and a traceable way, based on the article's setting, to calling $$f(x)=\cos(2\arccos(x))$$ a "polynomial function". I am still convinced that the given expression neither is a polynomial, nor even is strictly apt to yield a function (for the multivaluedness of arccos), but just evaluates in the mentioned interval for its principal values to the same values as the polynomial expression $$2x^2-1$$ does. I would enjoy an explanation for reverting my edit, which is amenable to me. Purgy (talk) 08:57, 7 June 2018 (UTC)


 * As usual, I have some difficulty understanding what you've written here. What our article actually says about polynomial functions is "A polynomial function is a function that can be defined by evaluating a polynomial."  The words "can be" are the reason that you are wrong.  The two different expressions (one involving a polynomial and one involving trig functions) are the same function: they have the same domain, same range, and same output on each input.  The multivaluedness is completely irrelevant: cos of arccos is not multivalued, even if we take arccos to be so.  --JBL (talk) 14:03, 7 June 2018 (UTC)
 * IMO, both formulations are confusing or misleading. The fact is that a function is not an expression, and an expression may define a function, but is never a function by itself. I'll try to clarify the sentence by merging the two formulations. D.Lazard (talk) 17:45, 7 June 2018 (UTC)
 * Before Purgy's edits, the word "expression" did not appear in the paragraph in question, and that was better. Now you have re-introduced it, which heightens (rather than avoids) the possibility of the misunderstanding.  To what advantage?  Also, the use of "clearly" is an extremely bad habit of professional mathematicians, whereas the phrase "does not look like" is both true and self-explanatory.  Finally, saying that a function "can be defined by a polynomial" removed the essential explanation of in what way it is defined by a polynomial.  I am reverting again.  --JBL (talk) 19:27, 7 June 2018 (UTC)

I submitted a new suggestion. Perhaps, it might be possible to improve its quality with more subtlety than by a simple revert. I am convinced that it contains an improved access.

Sadly, I cannot avoid to perceive the "difficulty to understand" as bordering a "can't hear you". In any case, I feel my initiating question as left unanswered, instead, wrongness being heaped upon me, without me having made a claim. Purgy (talk) 08:52, 8 June 2018 (UTC)


 * A polynomial is not an expression for exactly the same reason that a number is not a numeral. But just as Wikipedia would naturally say "the number 2" instead of the pedantic "the number represented by the numeral 2", so it is common sense to say "the polynomial represented by the expression" instead of something more technically accurate.  What?  "The set of ordered pairs whose first component is a number and whose second component is the number obtained when a polynomial expression is evaluated at the number." No!  There are situations where it is important to distinguish between the function and the expression representing the function as when, for example, we want to prove that there are uncountably many functions in a certain context, while the number of (finite) expressions on a finite set of symbols is clearly countable.  But an introductory article is not the place to raise this distinction.  Purgy's edit is ok. Rick Norwood (talk) 09:48, 8 June 2018 (UTC)


 * Purgy, your first edit contains no question marks (they look like this: "?"). So maybe you should not be surprised that I cannot see where you asked a question.  Your first edit was wrong (it introduced factually incorrect statements) and that is what I meant by calling it "wrong".
 * About the second version of Purgy's edit, I also do not think it is an improvement: it takes several different simple things and makes them more complicated. For example, why polynomials now are usually complex but can be restricted to the reals?  For an enormous number of people (e.g., essentially all students taking courses in calculus or below in the US, who should be able to understand what is a polynomial and what is a polynomial function), a polynomial has real coefficients and values unless specified otherwise, and this new description will be much less accessible.  There are other instances as well.  Since these edits are controversial and change many things each time, I would greatly prefer that Purgy self-revert and propose the individual parts one-by-one on the talk page for discussion.  --JBL (talk) 12:28, 8 June 2018 (UTC)
 * First Purgy's post does not contain question marks because none is needed when a question is set under the form "I would enjoy an explanation for ...". I do not see any mathematical error nor "factually incorrect statements" in Purgy's edits and in Purgy's posts. On the other hand there are several mathematical errors in your posts. For example The two different expressions (one involving a polynomial and one involving trig functions) are the same function: an expression is not a function, even if many expressions are used for defining functions. The confusion between expressions and functions is common among beginners in mathematics, but this is not a reason for introducing this confusion in Wikipedia. Also, it is wrong that the two expressions define the same function, as two functions with different domains cannot be the same function.
 * By the way, I have reread the present version of the article and done some small changes. Mainly, "evaluating" was linked to an article that does not contain the term. Also, I have changed "polynomial expression" into "polynomial" (I recall that, in this article, a polynomial is defined as an expression of a certain type). I have also changed "... which takes the values of the polynomial ..." into "... which takes the same values as the polynomial ...". After these edits, the section appears to me as both mathematically correct and not confusing. Thus, it should be kept. D.Lazard (talk) 14:04, 8 June 2018 (UTC)
 * JBL, it arguably belongs to the (hardship of understanding) to interpret (me being taken by surprise) and (enjoying an explanation) as a question, if not even as (an application for explanation), but I hope D.Lazard's pleadings and my parens are of good use in this hindsight. Anyway, thanks for the kind hint to the looks of a question mark, I'll test it below.
 * I also do not know which statement of mine in my first edit you classified as wrong. I just hold one vague suspect, for which I insist that the expression $$\cos(2\arccos(x))$$ is not strictly well formed, but only plausibly interpretable. The mathematical "can be" should not work across syntactical flaws, but as a logical quantor, which I tried to exploit in my second, still disliked, edit.
 * As regards the "in general" conception of polynomial functions in my new version re their coefficients, variables and values, may I ask to compare my version to the status quo ante? I claim that I, en passant, simplified the previous wording. Of course, I do not deny the possibility of further improvement. Go on, be bold in improving, not only in reverting.


 * Please, kindly specify more explicitly any other detriments you perceive. Purgy (talk) 16:12, 8 June 2018 (UTC)