Talk:Polynomial

Suggestion for a rewrite.
It seems to me that this article gets into advanced topics too quickly. They belong in the article, certainly, but perhaps in a later section, with the first part of the article covering the elementary properties of polynomials. What do you think? Rick Norwood (talk) 16:06, 27 February 2019 (UTC)
 * Could you be more specific? Looking to the article, it appears that sections "Definition" and "Arithmetic" would deserve to be moved just after "Etymology", as containing the basic tools that everyone should know about polynomial. The section "polynomial function" would probably deserve to be also moved up, although this could be the object of a discussion. Sections "Terminology" and "Classification" contain useful facts, which may be less important for beginners. Some parts of them would be better placed in "Definition" and "Arithmetic", and they can certainly be improved.
 * This being said, I find everything before the section "Solving equation" very elementary. What are the advanced topics that you have in mind? D.Lazard (talk) 17:24, 27 February 2019 (UTC)


 * Interpreting your recent edits I can imagine that your request might be roughly satisfied by placing content of the section "polynomial function" more at the beginning?
 * For the time being, I consider polynomials as abstract algebraic objects that employ formal powers of an indeterminate (mostly one). This makes them interesting on their own, without providing a domain for the indeterminates, changing the indeterminate to a variable, and requesting an extra homomorphism to allow for substitution of powers of elements from the domain, giving access to solving for roots, ... It is this distinction why not only I assign "indeterminate" preference to "variable" when talking about the extra symbol used in polynomials.
 * It is also in this context, where generally uppercase (X, Y, ...) letters are preferred for the abstract indeterminate, and lowercase (x, t, ...) letters for variables.
 * I understand that this fine distinction, and also the importance of the unsubstituted polynomials, may not be available in full depth, when reading the article for the first time, but I am unsure, if it is better to rub in the number affine polynomial function first, and only afterwards try to graft the abstract construction on the cured numerical routine.
 * Just my 2cents. Purgy (talk) 18:29, 27 February 2019 (UTC)

The section that bothered me was "Notation and Terminology", which goes on at some length about the (important) distinction between a polynomial, a polynomial equation, and a polynomial function. I think this could be stated in language more accessable to a non-mathematician.Rick Norwood (talk) 21:32, 27 February 2019 (UTC)

Revert of the new section
I have reverted the new section "Other bases" because; Nevertheless a section on positional notation, could be useful for saying that this notation represents a polynomial in the basis with some constraints on the coefficients. It would also be useful to explain that the arithmetic operations in positional notations are the polynomial operations, except for the carries. Please, try a better version. D.Lazard (talk) 20:32, 4 May 2019 (UTC)
 * As the term "polynomial" is not used, the relationship with the subject of the article may be unclear for a non-mathematician.
 * The heading and the main article should be Positional notation


 * I have rewritten the section somewhat, per your recommendations. Feel free to modify.  I did not insert material about the arithmetic operations.  Feel free to add that as you like.—Anita5192 (talk) 21:06, 4 May 2019 (UTC)

Edit to Arithmetic section
I've long felt that our mathematics articles need some improvements to make them more accessible to the layperson without giving up any mathematical rigor. While I felt this I haven't done anything about it until now. I'm going to try with the very baby step. I think this article generally starts out great, but when I get to one section I thought it could be improved. I won't write out my entire rationale here, but it may be worth glancing at User:Sphilbrick/Mathematics articles if you don't think my proposal is an improvement. —Preceding undated comment added 20:59, 17 July 2020 (UTC)


 * While I appreciate your intentions, I don't think it's very helpful to elaborate on such a trivial topic as polynomial addition. Seeing as you have a degree in mathematics, perhaps you could focus more on the more intricate passages of the text itself rather than simple formulas which are quite easy to follow. A bit further in the Arithmetic section, the discussion of division could use some improvement, e.g. there is no explanation of the last formula for complex numbers: $$5(x - 1)\left(x + \frac{1 + i\sqrt{3}}{2}\right)\left(x + \frac{1 - i\sqrt{3}}{2}\right)$$. This is arguably much less accessible to the layperson than the addition example as it was before. --147.229.196.135 (talk) 01:01, 18 July 2020 (UTC)
 * Sphilbrick and I have been chatting about this a little at my talk page. To the IP user, I think the key questions elided by your comment are "trivial to whom?" and "helpful to whom?"  Some part of the audience of this article should be students in what I would call "middle school" (say, ages 11--13); for the article to be helpful/usable to such a reader, I agree with Sphilbrick that, at least in the first section where basic arithmetic of polynomials is introduced, it would be good if the examples were spelled out in sufficient detail to illustrate the behaviors being described (e.g., that regrouping and reordering and simplifying all happen).   You are really right about the discussion of division and factoring; but there are plenty of things wrong with the section before one even gets to that part.  You are invited to pitch in as well :).  --JBL (talk) 01:27, 18 July 2020 (UTC)
 * As JBL mentioned, we've been chatting about mathematics articles. I don't disagree that the specific formula for complex numbers is less accessible to the layperson than polynomial addition, but I'm in a let's walk before we run mode. We also have to think about our audience. what's the over under for the percentage of adults who could and two polynomials. if someone says 50%, I want the under. I also want to distinguish between the proportion of adults who could answer the test question without studying (in my opinion south of 50%) and the proportion of adults who could follow a well-written example, and reproduce it on their own shortly thereafter (north of 80%). I see value in adding intermediate steps to help the reader. I am aware of WP:NOTTEXTBOOK, and will take care, but I don't think I've crossed that line. S Philbrick  (Talk)  14:49, 18 July 2020 (UTC)

Discussion of Polynomial
I like this section. I would characterize it as very good but not excellent. It has one shortcoming that I think is easily resolved.

I like the progression of the left side of the page starting with the simplest polynomial (degree zero), including the special case of the x-axis and the more general case, followed by degree 1, 2 etc. Then showing the general case of degree n. I like the parallelism of having formulaic expressions on the left side, and nice looking graphs on the right side.

I have two concerns, one of which is almost trivial: I'm trying to decide whether I should be troubled by having formulas on the left for degrees 0,1,2,3,n, while graphs for degrees 2,3,4,5,6,7. I've expressed the desire to add graphs for degrees 0 and 1, should I be troubled that we have a graph for degree for 4,5,6 and 7 but no formula? I'm not troubled but it may be worth discussing.
 * 1) why do we have the phrasing "zero polynomial" and "degrees 0 polynomial". not a big deal but slightly confusing. Should degrees be expressed as words or numbers? If both are acceptable maybe we should say so, but mixing numbers and words without explanation is suboptimal
 * 2) there's a nice graph for polynomials of degrees 2 through 7, but none for degrees 0 or 1.

If others concur, I'll try reaching out to the editor who created the graphics. I think it would be easy to create similar graphics for degrees 0 and 1. if those editors aren't responsive I'll try the graphics lab.-- S Philbrick (Talk)  14:36, 18 July 2020 (UTC)
 * About '"zero polynomial" and "degrees 0 polynomial"': Earlier in the article we have "The polynomial 0, which may be considered to have no terms at all, is called the zero polynomial". One cannot write "0 polynomial" as this is nowere defined. We could write the "the polynomial 0" instead of "the zero polynomial". But this could introduce a confusion with the degree, which is not zero in this case, but is indefinite. As all degrees that occur here are less than ten, we could use words instead of digit for degrees. This is recommended somewhere by the manual of style. In my opinion, keeping digits makes the classification more visible, and makes clearer the fact zero plays a different role in the two first items. So, my preference is the present version, but my preference is not a rule. D.Lazard (talk) 16:18, 18 July 2020 (UTC)
 * I have added graphs of degrees zero and one (those with the closest style that I have found in Commons). IMO, graphs of degrees 6 and 7 should be removed. 16:48, 18 July 2020 (UTC)
 * (EC) The difference in names might reflect that the zero polynomial is a proper noun, referring to the unique polynomial with that name. "degree 0 polynomial" is a noun phrase, with "degree 0" being an adjective modifying the noun "polynomial". It would certainly be odd to talk of the "0 polynomial". Using "degree zero polynomial" is less problematic but goes against convention for terse mathematical notation. --Salix alba (talk): 17:06, 18 July 2020 (UTC)
 * I agree with D.Lazard and Salix alba. To restate my previous comment, your effort would me more useful if you focused on the real issues instead of bike-shed discussions. --147.229.196.135 (talk)
 * , Thanks to D.Lazard for adding the graphs for degree zero and degree one. It didn't even occur to me to go check to see if there were suitable examples already. I concur that degree six and degree seven is overkill, not strongly enough to take them out myself but I wouldn't object if they were removed. Excellent point about the fact that the zero polynomial is not a polynomial of degree zero. I was so focused on the lack of parallelism I missed that point. I am aware of the general guidance that digits less than 10 should be spelled out, but I think that applies mostly to articles where the focus is on an issue other than math. I think there is value in using the numerals in this article. S Philbrick  (Talk)  14:37, 19 July 2020 (UTC)

I'm happy to see the addition of graphs for the degree 0 and degree 1 polynomials. At the risk of being anal, we now have four different styles of graphs, obviously because the graphs were created by different people with somewhat different styles.

As a separate issue, the formulas chosen are obviously different from one another, but more so than necessary (unless this is deliberate and I'm missing the reason why). I think that it would be good practice to start with something basic:

f(x)=2

Then keep adding terms:

f(x)=3x+2

Then multiply by, say x-2:

f(x)= 3x^2 -4x -4
 * = (3x+2)(x-2)

Possibly adding a scaling value if we want to keep the values within a containment region. Continue mutatis mutandes.

If this makes sense, looking to someone who can help us create the graphs.

Arithmetic – polynomial multiplication
I added two steps to the polynomial multiplication to show the intermediate steps.

I fully understand that people conversant with the subject will find the intermediate steps unnecessary, but many in our audience will find it helpful, and the more advanced we do can skip to the last step easily.

The rest of the section has some issues. It isn't that anything seems wrong, so it's hard to put my finger on it but it doesn't feel very organized. I'm in discussion with another editor about how it can be improved.-- S Philbrick (Talk)  14:27, 19 July 2020 (UTC)

Thus a sum of polynomials is always another polynomial
The arithmetic section contains a statement:

That statement immediately follows an example. The word "thus" in mathematics is typically a synonym for "therefore" and typically means that the result follows from the immediately preceding statements. I trusted it is obvious that a single example cannot provide proof of a universal claim (obviously, a single example can prove the falsity of a claim but that's not what is at issue here.) Arguably, the word "thus" doesn't simply refer back to the immediately preceding example, but refers to the opening statement of the section:

If that's the intention, then it should be moved up but we have other things to discuss.

Arguably, we can claim that while the wording is a little sloppy, the polynomials constitute a set, and it happens to be true that the set is closed with respect to the operations of " addition, subtraction, multiplication, and non-negative integer exponents of variables." I think that's true (I confess I'm rusty), but such a statement requires a reference. There is a reference at the end of the sentence: Polynomials The reference suggests that pages one and two are relevant, which they are, but I reread them twice and don't see anything that makes the claim about the set being closed to those operations.

Additionally, if the set of polynomials is closed to the operations of " addition, subtraction, multiplication, and non-negative integer exponents of variables." we ought to make that statement, with a reference, not simply the statement that (closure exists) with respect to addition.

At a minimum, we need a reference supporting a claim, and separately we need to decide whether to make the claim narrowly about addition or more broadly. Plus, as noted we either need to remove the word "thus" if we want the statement to immediately follow an example, or we need to place it properly in the section.-- S Philbrick (Talk)  18:58, 20 July 2020 (UTC)


 * You're right about "thus" (which was my addition while un-bulleting). Options are moving it up (but then the example is delayed, which I think is unhelpful), leaving it be (and hoping readers understand that it was supposed to refer back to before the example), or replacing the word "thus" with something else, like "In general" or "As in the example" or simply striking it (although I think that's awkward). --JBL (talk) 21:32, 20 July 2020 (UTC)

I'm trying to keep the terminology as simple as possible. It is understandable that some will feel that saying: "a sum of polynomials is always another polynomial " is simpler than saying: "the set of polynomials is closed with respect to the operation addition"

However, the term " closed" is the mathematical way of making the statement, and there are many many references supporting the claim that polynomials are close with respect to addition (because that's the way it is said) but it's harder to find references to make the arguably simpler statement. For that reason, I introduce the notion of closure, which permitted the addition of a recent reference. — Preceding unsigned comment added by Sphilbrick (talk • contribs) 18:06, 26 July 2020 (UTC)
 * It is wrong to use "closed" here, as "closed" is meaningful only for the restriction of an operation to a subset, and, here, the addition in not the restriction of an operation defined on a larger set. You can verify that the word "closed" does not appear in Operation (mathematics). Moreover, your edit does not follow the manual of style. So, I'll revert your edit. D.Lazard (talk) 19:55, 26 July 2020 (UTC)
 * , I'm interested in trying to improve mathematics articles. One common complaint is that they are not easily accessible to laypeople, and I hope to work on that but that issue is tangential to this specific issue. I've also noticed that a number of mathematical articles are under-referenced (obviously a more general problem across all our articles). I spend a lot of time at OTRS, where I field inquiries from the general public. (That's one source of the complaints about the relative level of the mathematical articles.) Another issue commonly raised there relates to whether an encyclopedia which can be edited by anyone can be relied upon. My stock answer is a variation of "it cannot. But if you see a statement that's of interest to you, go to the end of the sentence or the paragraph and you will probably find a link to a published reliable source which you may be able to rely upon". Given my stock answer, I thought I'd try to contribute to the solution by digging up references to some of the claims in mathematical articles.
 * The sentence in question:
 * Does have a reference after it. However, there is also a hidden comment which I will display here:
 * I think that comment is spot on. A reference doesn't have to have the exact wording of the Wikipedia article, but a layperson ought to be able to read the reference and conclude that the sentence in Wikipedia is correct. I read it twice and don't think it's all that obvious.
 * For that reason, my initial goal was to find a better reference to support the claim. When I searched, I could find plenty of textbooks asserting that the set of polynomials is closed under addition, but I was struggling to find one that would say that a sum of polynomials is always another polynomial or something very close to that. After thinking about it, I felt I knew why. Mathematicians typically don't say things like a sum of X is another X, they are more apt to say that a set is closed with respect to a particular operation. I decided to take the mountain to Mohammed. If I couldn't find a good source supporting this exact statement, I thought it would be appropriate to introduce the notion of closure, identify that formal mathematical concept is being the same as the more casual "sum of X is another X" and then include a reference that specifically speaks to closure.
 * You've objected to that construction. I frankly do not follow:
 * My guess is "in" should be "is" but even if that changes made, I'm not quite following. You also said:
 * I don't see why that's relevant. That article does talk about the domain of an operation and the range of an operation. That article doesn't go on to talk about situations where the range and domain match, but that's simply because it's beyond the scope of that article. Just off the top my head, that's roughly what we mean by closed — that the range and domain match. However, my point is that when we say that the sum of polynomials is another polynomial, we are saying that the set is closed with respect to the operations of addition. It seems like you dispute that (or something) so can we discuss where we disagree?
 * I see you also have concerns about the manual of style but let's resolve the underlying facts and then we can separately decide how best to present them.
 * As an important aside, my narrow concern is making sure the claim is properly referenced. If you can track down a reference that suppots the cirrent wording, then I'm not wedded to bringing in the concept of closure into this basic article. If you can't find a reference, I hope you will work with me so that we can rewrite this in a way that it can be properly referenced. S Philbrick  (Talk)  21:32, 26 July 2020 (UTC)
 * , Just repinging, on the chance you missed this. S Philbrick  (Talk)  12:56, 29 July 2020 (UTC)
 * The correct use of "closed" can be discussed, but it is not important here, as it is a technical term that is too technical here. If someone is not able to understand "the addition of two polynomials results in another polynomial", it will definively not understand the sentence "the set of polynomials is closed under addition", which involves two technical words ("set" and "closed") that are not useful here. IMO the use of the this phrase in this article is pure pedantry. I always wonder that many people think that using pedantic formulations helps the layman to understand mathematics.
 * About referencing. The whole content of section "Arithmetic" appears in hundreds of textbooks. So, I cannot understand why you have difficulties for sourcing that the sum of two polynomials is a polynomial. IMO, it would be worth to add, at the beginning of the section "Arithmetic" a sentence such that "Several operations are defined on polynomial, which are described in the following subsections", followed by a citation of a textbook that define polynomials similarly as here. This would make unnecessary all inline citations in subsections. D.Lazard (talk) 14:27, 29 July 2020 (UTC)
 * D.Lazard, I sincerely invite you to find a source that explicitly says that the sum of two polynomials is another polynomial, because all the texts I have consulted make this point implicitly, either via notation or via reference to an algebraic object (a vector space or a ring). --JBL (talk) 15:01, 29 July 2020 (UTC)
 * This is not a problem of sourcing, but a problem of wrtiting and logical coherency of the article. Presently, the section begins with "Polynomials can be added ...", which is a very poor definition, as the sentence suggests rather a property. A proper beginning would be: "the addition of polynomials is an operation that takes any two polynomials and produce always another polynomial, and is defined as follows." So, this is not the fact "the sum of two polynomials is a polynomial" that must be sourced, but the definition of the addition of polynomials.
 * By the way, it is also interesting to discuss what should be written after "at follows", as this strongly depend on the chosen definition of polynomials. With that of the article, the sum of the polynomials $P$ and $Q$ is simply the expression $P + Q$. In this case, the example shows that, if the input polynomials are in the normal form of linear combinations of monomials, then the result can be put in a similar normal form by using the method sketched on the example. If one defines a polynomial as a linear combination of monomials, the sketched method describes an algorithm for computing the sum. With other definitions of polynomials, other formulations must be used, but, formally, addition and multiplication should be defined by algorithms.
 * For a clear discussion of these questions, I would recommend the book Computer algebra by Davenport, Siret and Tournier. For better sourcing this article, I would suggest to choose one or two textbooks, to write the article in a way that most sentences are compatible with them, and search for sources only for sentences about facts that are not descibed in these books. D.Lazard (talk) 17:18, 29 July 2020 (UTC)
 * The section on addition of polynomials should be in principle understandable by a typical 12- or 13-year old. Layering the unnecessary logical structure of a definition including the technical term "operation" is not better in this regard than using vocabulary like "closed" (which I also am trying to avoid). Here is my view: one should ask "what should a person at level X get out of this section?"  If X is "someone who knows what is a vector space, or a ring, or ..." then the answer is "nothing, because they already know what polynomial addition is."  So that leaves the possible audience of X as, roughly, people whose knowledge of mathematics ends in secondary school.  In my opinion, what they should get out of this section is (1) polynomials can be added, (2) some amount of information about how polynomials actually are added, (3) the observation that the sum of two polynomials is another polynomial, and maybe also (4) an understanding of subtraction of polynomials.  Questions about sourcing etc. should be built around meeting those goals, not vice-versa. --JBL (talk) 16:53, 30 July 2020 (UTC)
 * , Both of you would like to avoid the term "closed". I'm with you, if we can find a solution. It might well be that your proposed formulation:
 * although it needs some word-smithing, and referencing. Making the assertion that the result of the operation produces another polynomial is a statement requiring a reference. It may seem obvious to the mathematically trained, but not to those who are not (our main audience) and it isn't trivially true; for example, the comparable statement about division is not true.
 * I left a request here, although I notice that few recent requests have not been fulfilled. (After rewording and finding a source, I'll cancel the request)  S Philbrick  (Talk)  13:25, 4 August 2020 (UTC)
 * Thanks to both of you.
 * I started with D. Lazard's wording, and decided to try:
 * To fit with the tone of the section. As a bonus, that phrasing resulted in a decent source covering more than addition but supporting the claim without the introduction of the word "closed". S Philbrick  (Talk)  13:52, 4 August 2020 (UTC)
 * About referencing. The whole content of section "Arithmetic" appears in hundreds of textbooks. So, I cannot understand why you have difficulties for sourcing that the sum of two polynomials is a polynomial. IMO, it would be worth to add, at the beginning of the section "Arithmetic" a sentence such that "Several operations are defined on polynomial, which are described in the following subsections", followed by a citation of a textbook that define polynomials similarly as here. This would make unnecessary all inline citations in subsections. D.Lazard (talk) 14:27, 29 July 2020 (UTC)
 * D.Lazard, I sincerely invite you to find a source that explicitly says that the sum of two polynomials is another polynomial, because all the texts I have consulted make this point implicitly, either via notation or via reference to an algebraic object (a vector space or a ring). --JBL (talk) 15:01, 29 July 2020 (UTC)
 * This is not a problem of sourcing, but a problem of wrtiting and logical coherency of the article. Presently, the section begins with "Polynomials can be added ...", which is a very poor definition, as the sentence suggests rather a property. A proper beginning would be: "the addition of polynomials is an operation that takes any two polynomials and produce always another polynomial, and is defined as follows." So, this is not the fact "the sum of two polynomials is a polynomial" that must be sourced, but the definition of the addition of polynomials.
 * By the way, it is also interesting to discuss what should be written after "at follows", as this strongly depend on the chosen definition of polynomials. With that of the article, the sum of the polynomials ⇭⇭⇭ and ⇭⇭⇭ is simply the expression ᙭᙭᙭. In this case, the example shows that, if the input polynomials are in the normal form of linear combinations of monomials, then the result can be put in a similar normal form by using the method sketched on the example. If one defines a polynomial as a linear combination of monomials, the sketched method describes an algorithm for computing the sum. With other definitions of polynomials, other formulations must be used, but, formally, addition and multiplication should be defined by algorithms.
 * For a clear discussion of these questions, I would recommend the book Computer algebra by Davenport, Siret and Tournier. For better sourcing this article, I would suggest to choose one or two textbooks, to write the article in a way that most sentences are compatible with them, and search for sources only for sentences about facts that are not descibed in these books. D.Lazard (talk) 17:18, 29 July 2020 (UTC)
 * The section on addition of polynomials should be in principle understandable by a typical 12- or 13-year old. Layering the unnecessary logical structure of a definition including the technical term "operation" is not better in this regard than using vocabulary like "closed" (which I also am trying to avoid). Here is my view: one should ask "what should a person at level X get out of this section?"  If X is "someone who knows what is a vector space, or a ring, or ..." then the answer is "nothing, because they already know what polynomial addition is."  So that leaves the possible audience of X as, roughly, people whose knowledge of mathematics ends in secondary school.  In my opinion, what they should get out of this section is (1) polynomials can be added, (2) some amount of information about how polynomials actually are added, (3) the observation that the sum of two polynomials is another polynomial, and maybe also (4) an understanding of subtraction of polynomials.  Questions about sourcing etc. should be built around meeting those goals, not vice-versa. --JBL (talk) 16:53, 30 July 2020 (UTC)
 * , Both of you would like to avoid the term "closed". I'm with you, if we can find a solution. It might well be that your proposed formulation:
 * although it needs some word-smithing, and referencing. Making the assertion that the result of the operation produces another polynomial is a statement requiring a reference. It may seem obvious to the mathematically trained, but not to those who are not (our main audience) and it isn't trivially true; for example, the comparable statement about division is not true.
 * I left a request here, although I notice that few recent requests have not been fulfilled. (After rewording and finding a source, I'll cancel the request)  S Philbrick  (Talk)  13:25, 4 August 2020 (UTC)
 * Thanks to both of you.
 * I started with D. Lazard's wording, and decided to try:
 * To fit with the tone of the section. As a bonus, that phrasing resulted in a decent source covering more than addition but supporting the claim without the introduction of the word "closed". S Philbrick  (Talk)  13:52, 4 August 2020 (UTC)
 * I started with D. Lazard's wording, and decided to try:
 * To fit with the tone of the section. As a bonus, that phrasing resulted in a decent source covering more than addition but supporting the claim without the introduction of the word "closed". S Philbrick  (Talk)  13:52, 4 August 2020 (UTC)
 * To fit with the tone of the section. As a bonus, that phrasing resulted in a decent source covering more than addition but supporting the claim without the introduction of the word "closed". S Philbrick  (Talk)  13:52, 4 August 2020 (UTC)

Summation Notation in Definition Section
I had made some edits which were reverted twice. It was about changing the Summation Notation in Definition Section from:
 * $$\sum_{k=0}^n a_k x^k$$

to
 * $$a_0 + \sum_{k=1}^n a_k x^k$$

The first one has a domain of all real numbers but 0 (with an indeterminate form 0^0 when x = 0), the second one has domain of all real numbers as is the case with polynomials. I understand that the former is simpler notation but since this is an important fundamental mathematical topic, I would argue that precision should be favoured over simplistic approach. --Niteshb in (talk) 01:17, 18 March 2021 (UTC)
 * I would have reverted it too. $$x^0=1$$ when the exponent is the integer 0. Without this meaning, rules like $$x^{i+1}=x\times x^i$$ don't work and zillions of other things break as well. Besides that, we follow reliable sources and they start the summation at 0 overwhelmingly. McKay (talk) 05:21, 18 March 2021 (UTC)

"A polynomial is not a function"?
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 $A$…Consider an infinite cyclic group generated by an element $X$. We let $S$ be the subset consisting of powers $X^r$ with $r \geq 0$. Then $S$ is a monoid. We define the set of polynomials $A[X]$ to be the set of functions (emphasis mine) $S \to A$ which are equal to $0$ except for a finite number of elements of $S$."

Thomas W. Hungerford gives a very similar construction:

"Theorem 5.1. Let $R$ be a ring and let $R[x]$ denote the set of all sequences of elements of $R\ (a_0,a_1,\ldots)$ such that $a_i = 0$ for all but a finite number of indices $i$. … The ring $R[x]$ of Theorem 5.1 is called the ring of polynomials over $R$."

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

"...a polynomial in one indeterminate is by definition a particular kind of sequence, that is, a function (emphasis mine) $\mathbb{N} \to R$."

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) $f$ defined for all numbers is called a polynomial if there exists numbers $a_0,a_1,\ldots,a_n$ such that for all numbers $x$ we have
 * $f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0.$"

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 $$a_0 + a_1x + \cdots + a_nx^n$$, where $$a_0,\ldots,a_n$$ are elements in an integral domain $$D$$ and $$x$$ is an element of an integral domain $$E$$ of which $$D$$ is a subdomain. They then define a polynomial function as one with a definition that can be written in polynomial form. 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 $$\mathbb N$$ or $$\{1, X, \ldots, X^n, \ldots\}$$ into the ring of coefficients, while your formulation means that $$X$$ 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. 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)


 * 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)


 * 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.
 * 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.
 * 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)
 * 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)

Imprecise definition of polynomial
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 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)

Proposal define special and normal polynomial.
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)