Talk:Polynomial

I came to this page looking for a quick reference on how polynomials are defined in contemporary mathematics. The "Definition" section didn't really have this information, and furthermore was barely sourced. I decided I should help out and improve it, and spent about twelve hours putting together a much more thoroughly-sourced version of that section covering polynomials in both an elementary context and in mathematics in general. My work was reverted within a day with little ceremony. I would be perfectly happy if people wanted to keep working on it, but it doesn't seem sensible to me to throw out all my hard work in favor of something much terser that isn't as grounded in high-quality sources. Much of this article would really benefit from more thorough citations and I was doing my best to help.

The reasons cited for the reversion was that "a polynomial is not a function" and the change needs consensus. So, here I am seeking consensus. Considering the objection that "a polynomial is not a function," here's Serge Lang on the matter:

We now give a systematic account of the basic definitions of polynomials over a commutative ring ${\displaystyle A}$…Consider an infinite cyclic group generated by an element ${\displaystyle X}$. We let ${\displaystyle S}$ be the subset consisting of powers ${\displaystyle X^{r}}$ with ${\displaystyle r\geq 0}$. Then ${\displaystyle S}$ is a monoid. We define the set of polynomials ${\displaystyle A[X]}$ to be the set of functions (emphasis mine) ${\displaystyle S\to A}$ which are equal to ${\displaystyle 0}$ except for a finite number of elements of ${\displaystyle S}$.^[1]: 23

Thomas W. Hungerford gives a very similar construction:

Theorem 5.1. Let ${\displaystyle R}$ be a ring and let ${\displaystyle R[x]}$ denote the set of all sequences of elements of ${\displaystyle R\ (a_{0},a_{1},\ldots )}$ such that ${\displaystyle a_{i}=0}$ for all but a finite number of indices ${\displaystyle i}$.

…

The ring ${\displaystyle R[x]}$ of Theorem 5.1 is called the ring of polynomials over ${\displaystyle R}$.^[2]: 149

Of course, a sequence is a function with domain ${\displaystyle \mathbb {N} ^{+}}$ or ${\displaystyle \mathbb {N} }$, as he notes soon after:

...a polynomial in one indeterminate is by definition a particular kind of sequence, that is, a function (emphasis mine) ${\displaystyle \mathbb {N} \to R}$.^[2]: 151

Lest anyone wants to claim that defining polynomials as functions isn't done in a more beginner-friendly context, Lang also defines them as such in his book Basic Mathematics, a pre-calc textbook appropriate for high school students:

A function (emphasis mine) ${\displaystyle f}$ defined for all numbers is called a polynomial if there exists numbers ${\displaystyle a_{0},a_{1},\ldots ,a_{n}}$ such that for all numbers ${\displaystyle x}$ we have

${\displaystyle f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x+a_{0}.}$^[3]: 318

Of course, this doesn't draw the same distinction between polynomials and polynomial functions that is made in more rigorous contexts, but that's why I described this style of definition separately.

It is true that other constructions are sometimes used, but I haven't seen any that are substantially different. For example, Birkhoff and Mac Lane in A Survey of Modern Algebra define a "polynomial form" as the form of an expression ${\displaystyle a_{0}+a_{1}x+\cdots +a_{n}x^{n}}$, where ${\displaystyle a_{0},\ldots ,a_{n}}$ are elements in an integral domain ${\displaystyle D}$ and ${\displaystyle x}$ is an element of an integral domain ${\displaystyle E}$ of which ${\displaystyle D}$ is a subdomain. They then define a polynomial function as one with a definition that can be written in polynomial form.^{[4]: 61–63} Although I did make use of their book, I didn't go into this construction because it wasn't present in any of the other sources I was using, it's mainly a semantic distinction, and I didn't want to make the definition section too long. After all, Birkhoff and Mac Lane don't rule out the idea that an "expression in polynomial form" itself describes a function distinct from a polynomial function—given their definition, what else would it describe? Even so, I'm happy to include their definition explicitly for the sake of thoroughness if other people here feel that's most important.

Anyway, I think this makes an open-and-shut case for the worth of the edit I made. This is material from popular textbooks written by luminary mathematicians on this topic, so it belongs in this article. As such, I'd like to restore my edit. I'd also like to do other work on this article—there are entire sections with no citations, which needs to be fixed.

Mesocarp (talk) 08:52, 5 September 2021 (UTC)

Firstly, what you wrote is a wrong interpretation of the sources: they essentially assert that a polynomial is a function with a finite number of nonzero values from ${\displaystyle \mathbb {N} }$ or ${\displaystyle \{1,X,\ldots ,X^{n},\ldots \}}$ into the ring of coefficients, while your formulation means that ${\displaystyle X}$ represents an element of the domain of the function, which is completely different.

Secondly the abstract definitions that you quote are counter-intuitive at elementary level, and this article is aimed to be elementary; see the hatnote. So, these definitions are not convenient here.

There are many possible definitions of polynomials, that are all equivalent, as said in Polynomial ring#Categorical characterization. The given definition is one of them, is elementary, and correct. So, it is best suited for this article. However, it should be said that there are other equivalent definitions (with links to Polynomial ring), and sources must be provided (there are sources that give this definition, even if I have not them under hand).

You are talking of "contemporary mathematics". It must be noted that the definition given here is based on the concept of expression. This concept arised in mathematics for the need of computer algebra, and was not considered by your sources. So the definition of the article is in some sense more contemporary than your sources. D.Lazard (talk) 10:56, 5 September 2021 (UTC)

I agree wholeheartedly with D.Lazard’s revert and would like to emphasize, in particular, that this article includes among its potential readers students in primary and secondary school, and it is absolutely essential that it be written to serve them, not just people with exposure to modern algebra. I second the suggestion that you seem to want to be editing the higher-level article polynomial ring. If you do go edit that article, I hope you will consider carefully how your edits fit in to the article as a whole —- that was unfortunately missing here. —JBL (talk) 11:12, 5 September 2021 (UTC)

The idea that this article should be written towards an audience of primary or secondary school students and other sorts of material left out strikes me as against WP:TECHNICAL, especially WP:OVERSIMPLIFY. Polynomials are discussed in a wide variety of reliable sources, including those aimed at primary or secondary school students, but also those aimed at undergraduate and graduate students, working mathematicians, students and specialists in other fields, and the general public. Ideally, this article should represent a synthesis of all of these per WP:BALASP. I don't think it's our place to guess about what primary or secondary school students may or may not find accessible and cherry-pick all our sources on that basis.

I started off my edit with a definition of polynomials sourced from a pre-calc textbook, so it's not like I'm ignoring grade-school-level material. I think that's more appropriate: to the extent that sources at that level and sources at a more specialized level tend to substantially differ, we ought to cover both perspectives. Otherwise we present a skewed view of the topic.

Polynomial ring doesn't cover the same subject as this article. Obviously they have a very close relationship, but I doubt there's a single reliable source out there that would say that a polynomial and a polynomial ring are exactly the same thing, nor do I get the impression that any of us think they are—that's why we have two different articles, after all. Putting all the specialist material in that article and all the elementary material in this article doesn't reflect how polynomials are treated in the sources we have available. On that basis, I think that hatnote needs to be removed and this article needs to be handled in a more balanced fashion instead: the hatnote is a symptom of the problem.

If we really thought it was necessary to have a separate article just for the elementary material, I think the appropriate way to do that would be through an introductory article. However, I'm not convinced at this point that the elementary sources and other sources diverge so widely as to make this necessary; it seems like a method of last resort.

I think it will be easier to wait on discussing issues of accuracy, article flow, or the like until after we've settled this point. After all, if most of the sources I cited are considered ineligible for use here, everything else kind of falls away. Mesocarp (talk) 19:51, 6 September 2021 (UTC)

The first sentence of WP:TECHNICAL is The content in articles in Wikipedia should be written as far as possible for the widest possible general audience. The number of students in grade 10 is several orders of magnitude larger than the number of working mathematicians, etc. The article at present contains a very nice discussion of polynomial arithmetic that is understandable by students in grade 10, and which is also understandable by working mathematicians (I co-wrote it, and I am one). Your edit placed a vastly more technical definition of these simple operations in an earlier section while leaving the elementary definition in place, lower in the article -- this breaks any principle of good writing and common sense.
There is plenty of room for improvement of this article, both in sourcing and content. A necessary starting point is that you should begin by carefully reading what is already in the article, by carefully reading what your interlocutors are writing, and by carefully reading the sources. (For example, it is not clear from your response whether you have understood that the function in the Hungerford definition is completely different from the function in your edit.) I am sure that we can find a more constructive way to re-start this conversation. --JBL (talk) 21:11, 6 September 2021 (UTC)

@Mesocarp: You may be interested in editing Indeterminate (variable). This article is currently needs additional citations.--SilverMatsu (talk) 03:10, 7 September 2021 (UTC)

Maybe so; I'll take a look. Mesocarp (talk) 08:36, 7 September 2021 (UTC)

The first sentence of WP:TECHNICAL is The content in articles in Wikipedia should be written as far as possible for the widest possible general audience.

That's not all it says, though. WP:TECH-CONTENT, on top of WP:OVERSIMPLIFY, strongly implies that specialized material belongs in this article—not at the expense of elementary material, but in addition to it. What elementary sources and specialized sources say on this topic is significantly different, as I think we both recognize.

Your edit placed a vastly more technical definition of these simple operations in an earlier section while leaving the elementary definition in place, lower in the article -- this breaks any principle of good writing and common sense.

The very first paragraph of my edit was sourced from a pre-calc textbook called Basic Mathematics. Maybe you think my summary of that material was too opaque and it should be made clearer, or that more space should have been given to material of that sort. That's fine, and you could certainly work it over to that effect. It doesn't imply that my edit should have been reverted.

To be honest, I didn't leave the existing material in at all; it didn't seem very thoroughly cited and I figured its contents would be preserved to the extent that they appeared in my sources. I admit that was a bit irresponsible of me. I feel a bit iffy about basing a whole section largely around Wolfram MathWorld because it's a tertiary source (see WP:TERTIARYUSE), but their article has a nice list of references at the bottom that we could try to bring in here, too. I'm more than happy to include the existing definition in full to the extent that we can source it well, and at the top of the section if you like.

…it is not clear from your response whether you have understood that the function in the Hungerford definition is completely different from the function in your edit.

Sure, I was working more from Lang. To me, Hungerford's definition seemed close enough to Lang's that I didn't feel the need to go into them both separately, but I acknowledge that substantial differences between them exist, and it sounds like you feel they're quite severe. So, why don't we talk about their differences explicitly in the article? Sounds great to me. Mesocarp (talk) 08:36, 7 September 2021 (UTC)

I am afraid that your response is sapping me of the sense that constructive engagement could work here, because you do not really seem to have read the article as written, nor gone back and compared your edit to Hungerford when prompted. I will disengage. —JBL (talk) 10:34, 7 September 2021 (UTC)

Well, that's certainly your right. I'll do my best to take your concerns into account in my further work on this article. If you have a problem with edits I make going forward, I really hope you'll reconsider your decision. Mesocarp (talk) 10:53, 7 September 2021 (UTC)

I've no idea what "a polynomial is not a function" is intended to mean, but the definition section should indeed be written at a secondary school level or lower (and much lower than pre-calc), and the changes made were undesirable. That polynomials in elementary algebra can be generalised to something also called "polynomial" in abstract algebra does not mean that the elementary definition is wrong. More complex information belongs later down the article. Consider it this way: either way, you've invested 12 hours in reading and internalising lots of sources about polynomials, and you can either spend time arguing about it in a discussion that won't go your way, or you can use that newfound knowledge to make improvements that we all agree should be made (sourcing unsourced content, improving more technical articles). — Bilorv (talk) 19:24, 7 September 2021 (UTC)

The definition of polynomial is more complex that needed, repeated twice (in introduction and not-so-formal definition) and not really well defined: "... polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables."

What is a coefficient and what is its role in the polynomial?. They are mentioned to exists and then not "used" in the definition. I think it is better to be pragmatical and start with an example, then try to make it clear what the operations allowed to variables and what is the meaning of "coefficients x (variable exponentiation)". What "a·x²" means in practice. "x²" is vector and "a" a transposed constact vector (sort of escalar product)? Not clear at all from the definition. Or maybe the coefficient are SU matrices and the result is of evaluation the polynomial is just another vector? Or maybe "a·x²" must just be interpreted as another new variable?

"Indeterminates" does not even appear in my English dictionary. "Variable" is well known and understood and is more "friendly" with terms used later in the classification of polynomial (univariate, multivariate, ...). "Indeterminates" distract attentions without providing any meaningful information. — Preceding unsigned comment added by 88.4.162.172 (talk) 10:46, 31 December 2021 (UTC)

Thank you for pointing imprecisions in the definition. I have edited § Definition for fixing them. I have not changed the lead, as, there, readibility is commonly preferred to accuracy. Now, "coefficient" and "constant" are defined.

"I think it is better to be pragmatical and start with an example": there are two examples in the first paragraph of the article.

I do not understand why you suppose that variables and coefficients are vectors and matrices. This is absolutely not the case.

Apparently your dictionary is not a mathematical dictionary. "Indeterminate" as a noun, is a standard term that is defined in the provided wikilink and also in the article. It is preferred to "variable" (the historical term) because "variable" suggest wrongly the idea of variation, that is not present here. Rather, "Indeterminate" means "unspecified". D.Lazard (talk) 11:51, 31 December 2021 (UTC)

I would like to add the definitions of a special polynomial i.e. greatest common denominator(P,DP)=P and normal polynomial i.e. greatest common denominator(P,DP)=1. This is important in understanding the algorithms for symbolic integration. Any concerns? TMM53 (talk) 09:17, 2 January 2023 (UTC)

This does not seem a good idea. I guess that you are talking of univariate polynomials, that DP means the derivative of P, and "greatest common denominator" means "greatest common divisor" (since there is no denominator in a polynomial). If my guesses are correct, what you call a normal polynomial is standardly called a square-free polynomial, and there is an article on this topic. If my guesses are correct, what you call a "special polynomial" is an inseparable polynomial, and exists only over a field of finite characteristic. So this is not useful for integration.

Also, you are talking of "the algorithm for symbolic integration". There are many such algorithms. I guess that you are talking of integration of rational functions. Even for this restricted case, there are several algorithms, and I guess that you have "partial fraction decomposition" in mind, which uses heavily square-free polynomials and square-free factorization.

In any case, adding new definitions here does not help to understand the existing articles on symbolic integration, especially if this introduces uncommon terminology. D.Lazard (talk) 11:10, 2 January 2023 (UTC)

Indeterminate is a term used almost exclusively in reference to rings of polynomials. The more accessible, and more general term is “variable”. Farkle Griffen (talk) 16:36, 22 August 2024 (UTC)

The term "indeterminate" long predates the concept of a ring. Here in 1715 x is called an "indeterminate quantity", and here in the 1840s we have some "unknown x" called "indeterminate" as an adjective and on a later page "indeterminate" is used as a noun). The ring theory people seem to have adopted the term (sometime in the 1920s or 30s maybe?) and given it a more precise definition in certain contexts, but it is still used more generally, as in this example from the 1970s or this example from 2014, or this other example from 2014. –jacobolus (t) 17:47, 22 August 2024 (UTC)

With that said though, I don't have a strong feeling about whether this article should use "unknown", "indeterminate", "variable", or some other synonym. –jacobolus (t) 18:48, 22 August 2024 (UTC)

"Indeterminate" as it is used in the first books is not used in the same sense as it is used in this article. In those books, the term is used as a property of unknowns in an equation (or in a geometric problem as in the first book) and moreover, in those books, the term used to define the variable in a polynomial is simply "letter", not "indeterminate". And I don't think you would argue that we should write this article declaring the variable symbol to be called "letter" simply because these older books do so.

But sure, the term "indeterminate" is more general than in ring theory. But the point still stands; "variable" is the more accessible term, and "indeterminate" has a defined meaning which this article incorrectly links to.

More formally, in ZFC and mathematical logic, formulas and expressions are defined in terms of variables, see Zermelo–Fraenkel set theory#Formal language and Expression (mathematics)#Formal definition and Term (logic). The term indeterminate is not formally defined, unless one is referring to the definition described in the Indeterminate (variable) article, wherein even that is defined by the more formal term "variable".

And just subjectively, I have never seen the term "Indeterminate" used in any context other than to define a ring over a type of expression. Going back through my algebra books, all of them either define polynomials in terms of variables, (see Lang), or if they use the term "indeterminate", they are specifically talking about the ring of polynomials (see Dummit and Foote). Farkle Griffen (talk) 19:28, 22 August 2024 (UTC)

The adjective "variable" in "variable quantity" originally (e.g. by Isaac Newton) applied to quantities which were explicitly changing, for instance over time, whereas the adjective "indeterminate" implies that a quantity cannot be determined (not that it is changing), and "unknown" implies that the quantity's value is not yet (but perhaps could be) known. These are all slightly different claims. Nowadays, these plain-English meanings have been trampled by jargon uses which conflate, ignore, or even contradict the plain meanings of words in favor of giving them narrow and context-dependent technical definitions, in ways which are often inconsistent or contradictory from one context to another. –jacobolus (t) 20:08, 22 August 2024 (UTC)

There's also a common confusion (in literature broadly and in Wikipedia) between a quantity and a symbol used to represent it, with names like "variable" sometimes applied to the quantity, and sometimes applied to the symbol, and with conventions about which is implied by the word varying from one topic or source to another. –jacobolus (t) 20:17, 22 August 2024 (UTC)

The point still stands. The article overuses and misuses the term indeterminate. "Variable" is more accessible, being taught all the way back to elementary algebra (which most readers of this are expected to be familiar with) so using indeterminate would only be more confusing for the reader; it is more general, since as a formal term, indeterminate refers to a specific kind of variable which is almost never what a non-mathematician would mean to when describing a polynomial; and more formal, since, the term "variable" actually has a formal definition that describes the symbol being used in the article perfectly well.

I see no reason why this article should rely so heavily on the term "indeterminate". Farkle Griffen (talk) 20:45, 22 August 2024 (UTC)

Isn't the whole point of a polynomial that you can put anything in for the x? It seems like this is adequately covered by the plain meaning of "indeterminate", but not necessarily of "variable". Variable should be reserved for the latter case, when one is thinking of the polynomial primarily as a function. Thus when one talks of variables, there is usually some specific domain at issue, e.g., a "real variable" or a "complex variable". In some situations, there is even an important distinction. For example, one might say that the polynomials ${\displaystyle x^{p}}$ and ${\displaystyle x}$ are the same for ${\displaystyle x}$ a variable taking values in the field with p elements (one probably should avoid saying this actually true statement, because of the high probability it would be misunderstood). On the other hand, the polynomials ${\displaystyle x^{p}}$ and ${\displaystyle x}$, modulo p, are not the same polynomial in an indeterminate x (modulo p). Tito Omburo (talk) 21:59, 22 August 2024 (UTC)

Yes, and that is exactly the problem. You cannot put anything in for an indeterminate. The algebraic structure of the set of polynomials in indeterminates is said to be isomorphic to rings of polynomials in variables with nonempty domains. More generally, a variable can represent an indeterminate (specifically the variable with an empty domain) but indeterminates cannot be substituted for anything because they represent absolutely nothing. Which is exactly the problem.

You're correct that defining the equality of polynomials over an arbitrary domain is not as simple as comparing their coefficients, but that is something that needs to be talked about in the article anyway. It would be just as, if not more confusing to say that “The two polynomials ${\displaystyle x^{p}}$ and ${\displaystyle x}$ are not equal modulo p”. Farkle Griffen (talk) 22:54, 22 August 2024 (UTC)

Also, as a side note, one could argue that the polynomial ${\displaystyle x^{p}}$ doesn't exist in the ring of polynomials mod p, since, by some definitions, those polynomials are only up to degree ${\displaystyle p-1}$, and multiplication is defined to be multiplication modulo p. However it's not inconceivable that one chooses to define the ring in such a way that a situation like the one you describe arises. Farkle Griffen (talk) 23:06, 22 August 2024 (UTC)

I think you miss my point, which is that the polynomial ${\displaystyle f(x)=x^{p}\in F_{p}[x]}$ in the indeterminate ${\displaystyle x}$ is not the same as the polynomial ${\displaystyle x\in F_{p}[x]}$, but when ${\displaystyle x}$ is a variable in ${\displaystyle F_{p}}$, these are exactly the same. Tito Omburo (talk) 00:32, 23 August 2024 (UTC)

Yes, I understand that. But that has to be mentioned either way. I don't see how that is justification for having the entire article rely so heavily on indeterminates. Farkle Griffen (talk) 01:26, 23 August 2024 (UTC)

It's generally a fair point. I don't like the article's definition of a polynomial as a mathematical expression, which is a very formal point of view common in higher mathematics, but is not universal and is somewhat specialized and inaccessible (and the article proceeds to ignore this definition at several later points).

The definitions cited in the several footnotes provided as sources for it also do not agree. For instance, McCoy has something similar, but is very explicit that its concept of a polynomial is a "purely formal expression. That is, the '+' signs are not to be considered as representing addition ..." Beauregard says "A polynomial over ⁠${\displaystyle F}$⁠ is a sequence of the form ⁠${\displaystyle f(x)=(a_{0},a_{1},a_{2},...)}$⁠ where each ⁠${\displaystyle a_{i}\in F}$⁠ and all but finitely many ⁠${\displaystyle a_{i}}$⁠ are zero." Moise defines a polynomial as a kind of continuous function.

It might be an improvement to lead by saying that a polynomial is a sum of terms each of which is a non-negative-integer powers of one or more variables without calling it an expression, a sequence, or a function up front, and then include a section explicitly describing the differences between these points of view. –jacobolus (t) 02:08, 23 August 2024 (UTC)

I think this is already explained in the article. Tito Omburo (talk) 02:35, 23 August 2024 (UTC)

It is to some extent, but the explanation could be a lot clearer, more direct, and more accessible in my opinion. I think we should be expecting people with no more than a middle school algebra level of knowledge to come read this article, so we should try to make the first 2–3 sections accessible to them. –jacobolus (t) 20:22, 23 August 2024 (UTC)

As a person who, in day-to-day practice as a professional mathematician, rarely uses the word "indeterminate" and does not distinguish it from "variable", I have been puzzled about whether your edits are improvements; one could argue that the polynomial ${\displaystyle x^{p}}$ doesn't exist in the ring of polynomials mod p is clarifying in that I now know for sure that you are very confused about some core aspects of this topic. Every polynomial (say with coefficients in a field) gives rise to a polynomial function (via substitution of its indeterminates); in positive characteristic this is not a bijective correspondence; and yet the distinctness of the polynomials ${\displaystyle x}$ and ${\displaystyle x^{p}}$ over ${\displaystyle \mathbb {F} _{p}}$ is extremely important and extremely well atteseted to in all sorts of excellent sources. Are your edits grounded at all in reliable sources? --JBL (talk) 18:58, 23 August 2024 (UTC)

You're right, I'm a bit in over my head here, but I do believe that the article should not be so heavily reliant on "indeterminate"; though the exact reason has changed. This started by trying to clarify the Indeterminate (variable) artice. I talked to the professors in my university's math department, and they (apparently) did not know a formal definition either (many of them were very adiment that indeterminates could not be substituded), but they eventually agreed that a formal definition was a “variable with no inherent domain”. So this was I focused the article on that, with the idea of finding more sources later, coming up with this version: https://en.wikipedia.org/w/index.php?title=Indeterminate_(variable)&oldid=1241754252

But this was (rightly) reverted citing WP:Original Research, as I could not find sources in time. Then, in my attempt to find formal definitions, I could only find three books which attempt to formally define the term which define it (very surprisingly) as I describe in this version: https://en.wikipedia.org/w/index.php?title=Indeterminate_(variable)&oldid=1241888326

(You may see the books in this version of the article) Almost word-for-word, and all of them agreed on this definition. (please read the sections of the books before you just tell me it is wrong). And because it was so different from how it is usually presented, I started a discussion in the talk page, here: https://en.wikipedia.org/wiki/Talk:Indeterminate_(variable)#Disparity_between_definition_and_usage

Which also has links to exact page numbers. However this version was also reverted, and is currently under discussion in the Talk page over there. Farkle Griffen (talk) 19:43, 23 August 2024 (UTC)

However, in any case, the exact definition of "indeterminate" seems to be more complicated and and more strict than simply "variable". The article should not rely so heavily on this term; "variable" should be preferred for the reasons I’ve mentioned above. Farkle Griffen (talk) 19:56, 23 August 2024 (UTC)

It is true that some textbooks on abstract algebra define an indeterminate over a ring R as an element of a larger ring that is transcendental over R. But, such a definition that is related to a base ring is not used outside elementary abstract algebra. In particular no number theorist would consider π and e as indeterminates. Nevertheless, I have mentioned this other definition of an indeterminate in Indeterminate (variable). D.Lazard (talk) 10:40, 24 August 2024 (UTC)

Comments:

• I agree that "variable" can be used as a synonym of "indeterminate" in the case of polynomials with numerical coefficients. When the coefficients depend on variables, the use of "variable" instead of "indeterminate" can be confusing. For example, the general quadratic polynomial ⁠${\displaystyle ax^{2}+bx+c}$⁠ involves a single indeterminate (x and 4 variables, including the indeterminate).
• ""indeterminate" has a defined meaning which this article incorrectly links to": The link was incorrect because Farkle Griffen changed the target article by introducing another definition of an indeterminate, which is own original resarch replacing the previous definition with another one that is not used outside elementaty abstract algebra. I have reverted this change in Indeterminate (variable), and it results that there is no more misuse.

D.Lazard (talk) 11:41, 23 August 2024 (UTC)

The redirect Simple root (polynomial) has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2024 September 7 § Simple root (polynomial) until a consensus is reached. 1234qwer1234qwer4 16:20, 7 September 2024 (UTC)

Since UNSW Professor Normal Wildberger released a paper with the solution to the general polynomial, is there a need to change some of this?

https://www.tandfonline.com/doi/full/10.1080/00029890.2025.2460966 — Preceding unsigned comment added by Cthulhubc (talk • contribs) 08:07, 3 June 2025 (UTC)

The answer is no, since this is an original research that has not been discussed in any other mathematical paper. The Wikipedia basic policy WP:No original research forbids the use of such a paper that has not beem discussed in an independent secondary source. D.Lazard (talk) 08:56, 3 June 2025 (UTC)

It's not accurate to say that Wikipedia policy forbids the discussion or citation of research papers published in The American Mathematical Monthly. However, adding significant discussion of new research sometimes gives a misleading impression of a field; where possible it's better to find a survey paper or textbook as a source, as these typically provide a more balanced and comprehensive summary of a topic. See Wikipedia:Neutral point of view § Due and undue weight. –jacobolus (t) 13:59, 3 June 2025 (UTC)

It is true that many articles of The American Mathematical Monthly are useful as reliable secondary or tertiary sources. In paticular, the history section of this article could be used as a reliable source for its content. But, otherwise, this aticle is clealy assumed to be original research and, so, has to be considered as such, and must not be included in Wikipedia before having secondary sources establishing the notability of the results (the author and the journal are notable, but this is not sufficient for establishing the notability of the result). D.Lazard (talk) 14:59, 3 June 2025 (UTC)

Also relevant are Wikipedia:Articles for deletion/Norman Wildberger and Wikipedia:Articles for deletion/Catalan Geode. --JBL (talk) 00:15, 4 June 2025 (UTC)

Wildberger is effective at pissing off some prickly mathematicians by (a) having a different philosophical point of view than the mainstream and writing about it polemically, (b) making somewhat sensational speculative claims and then hyping them to the popular press, (c) doing a lot of self-citation and not always doing a good job citing others' work, even in his papers that are otherwise uncontroversial, not to mention making up weird personal names for things, and (d) attracting a large YouTube audience, many of whom are non-experts and some of whom try to promote Wildberger's work out of proportion in places like Wikipedia. He still seems notable as a person (e.g. as a YouTube personality and polemicist), if someone is willing to write a careful neutral article. The discussion at Wikipedia:Articles for deletion/Norman Wildberger is not, itself, particularly neutral and I think some Wikipedians let their personal feelings about Wildberger get the better of them. –jacobolus (t) 01:11, 4 June 2025 (UTC)