# Talk:Polynomial

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## 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)

Done. 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 (talkcontribs) 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 (talkcontribs) 23:56, 27 December 2013 (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 (talkcontribs) 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)

## 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 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)