Talk:Polynomial/Archive 2

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Archive This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page.

Contents

Intro paragraph

Oleg Alexandrov warns in an edit summary that the introduction is becoming confusing. This may be right; as the editor who added the most recent bit of confusion I feel somewhat responsible. Perhaps it could be simplified by starting with polynomials in one variable (which is what a lay reader will be most likely to have encountered) and only generalizing to multiple variables later? –Henning Makholm 23:46, 3 March 2007 (UTC)

If one reads the whole article, one will see that this article is about polynomials in one single variable, except a single section at the bottom which is about polynomials in several variables. That's how it should be, and the intro should reflect that.
I think the intro should be very short, concentrating on polynomials in one variable, then mentioning in at most one paragraph that polynomials can be considered in more than one variable too, and that this will be discussed later in the article in more detail. Oleg Alexandrov (talk) 03:53, 4 March 2007 (UTC)

How about this?

In mathematics, a polynomial is an expression of the shape

a_nx^n + a_{n-1}x^{n-1} + \cdots + a_2x^2 + a_1x + a_0

where the ai's are constant numbers, x is a variable. The integer n\geq 0 is often chosen such that an is nonzero; in that case n is the degree of the polynomial. If all of the ai's are 0, the polynomial is the zero polynomial; it is usually not assigned a degree.

There may be minor variations in the typographical form of a polynomial; for example the expression

3 x^4 + \tfrac{x^3}{2} - x - 12

is a generally accepted shorthand notation for the polynomial

3 x^4 + 0.5x^3 + 0 x^2 + (-1)x + (-12) \,.

More generally, it is often convenient to let the word polynomial mean any expression that contains only constants, variables, addition, subtraction, and multiplication, such as

(x^2-2)(\tfrac{1}{2}x(1+6x)+6)

This meaning differs only superficially from the one given above, because the rules of elementary algebra can always be used to rewrite such an expression to be transformed into the more restricted shape.

A polynomial in several variables may contain multiple variables rather than just x. Its canonical form is a sum of terms of the shape a_{ij...k}x^iy^j\cdots z^k.

(Some consequential changes to the "elementary properties" section will also be necessary. –Henning Makholm 01:35, 11 March 2007 (UTC))

Polynomials are defined by recursion

The definition of a polynomial is the following three conditions.

  1. Any number is a polynomial.
  2. The symbol X is a polynomial.
  3. If a and b are polynomials, then so is the sum a+b and the product a·b.

So, examples of polynomials include:

  1. The zero polynomial 0,
  2. A constant polynomial 3,
  3. The variable X,
  4. A first degree polynomial 3+X,
  5. A third degree polynomial (3+X)·(2+X·X) = X3+3·X2+2·X+3.

Bo Jacoby 14:29, 11 March 2007 (UTC).

I don't like this recursive definition. It is simpler to say that polynomials are obtained using addition and multiplication or specifically write out how a polynomial looks like than define things recursively. Oleg Alexandrov (talk) 20:20, 11 March 2007 (UTC)
Seconded. Recursive definitions are good for rigid reasoning, but not good for encyclopedic introductions, which ought to be helpful even for mathematically unsophisticated readers who tend to be confused by recursion. Starting out with a recursive definition is almost always the wrong thing to do in an enclyclopedia article, even though it may well be the right thing to do in a research paper or a set of lecture notes. –Henning Makholm 22:15, 12 March 2007 (UTC)
I support this definition, with a change to the 2nd bullet. Rather than saying X is a polynomial, it should be generalized to "a variable is a polynomial". This would account for polynomials of many variables. Never before have I seen such a clear, concise, and accurate definition. —The preceding unsigned comment was added by Eat2thepieseye (talk • contribs) 11 June 2007.
Thanks for the support. However, the concept of a variable is tricky. From the point of view of analysis, the argument x of a polynomial f(x), is a variable, because various constants may be substituted for x. From the point of view of algebra, the polynomial f is a member of a polynomial ring, and so is the polynomial x . So it is a constant. In the polynomial function (x \mapsto f(x)), x is the variable and f is the constant, while in the homomorphism (f \mapsto f(x)), f is the variable and x is the constant. See what I mean? Bo Jacoby 10:09, 11 June 2007 (UTC).
I think it's a problem that you don't define the product of two polynomials, but you use the product in the definition. Is it associative? commutative? does it have a distributive law? Are there any relations like X*X = 0 (see dual numbers). This "definition" fails to be a definition at all. If you want a rigorous definition, you probably have to be content with something along these lines:
A polynomial ring (K[x], α, x) consists of a ring K[x] with a distinguished element x ∈ K[x] and homomorphism α : K → K[x] such that the following universal property holds: for any triple (R, β, r) where R is a ring, β : K → R is a homomorphism, and r is a distinguished element of R, there exists a unique ring homomorphism φ : K[x] → R such that
\phi(x) = r,\quad \beta\circ\phi = \phi\circ\alpha
Then, of course, a polynomial is just an element of a polynomial ring. Silly rabbit 13:17, 11 June 2007 (UTC)

Height

Can someone please include a brief description of the "polynomial height"? I recently came across this concept while studying for the GRE math subject test and I have not been able to find a good explanation. Thank you. 24.193.160.55 01:28, 12 March 2007 (UTC)

It sounds as if the question was badly worded or else you did not read it carefully. There is a difference between a polynomial, a polynomial equation, and a polynomial function. Only the latter has a well defined height. Height of a polynomial function is the y-coordinate for a given x-coordinate. For example, the height of y = x^2 at 5 is 25. Rick Norwood 19:39, 12 March 2007 (UTC)
I don't think so, Rick. The height of a polynomial is defined to be the maximum magnitude of the coefficients of the polynomial. See [[1]]. This ought to be on wikipedia somewhere - I'll add it when I get a chance. Cheers, Doctormatt 19:50, 12 March 2007 (UTC)

You are correct, though I have never run into this before. Is it really something they would ask on the GRE? Rick Norwood 21:15, 12 March 2007 (UTC)

It is the math subject test, so I think it is possible. Heights of polynomials and related topics appear in algebraic geometry and number theory, among other places. - Cheers, Doctormatt 21:32, 12 March 2007 (UTC)
Height of a polynomial now exists. Cheers, Doctormatt 06:33, 13 March 2007 (UTC)

What in the heck?

Has this article been recently vandalized? What is with the intro? It was either written by someone with some sort of high-level structure dyslexia, or was rearranged by a vandal. Brentt 22:26, 19 March 2007 (UTC)

I don't see evidence that the article has been vandalized (except a few blatant vandals earlier today, which were reverted quickly). Its current sorry state appears to be the result of several people (of which I am one) editing it in good faith, but based on different ideas about which definitions should be basic in the article. Unfortunately this has left the article woefully inconsistent, and it is in dire need of a complete rewrite/reorganization. I hope to get time to do this Real Soon Now, but would be delighted to be beaten to it.
Currently I think that the article ought to first give a solid account of real/complex polynomials in one variable, downplaying the distinction between a formal polynomial and a polynomial function, which is not essential in the real/complex case. This will cover most of the substance in the current article except the initial attempts to give a final definition first. Later sections should then introduce generalizations to polynomials in multiple variables, polynomials over arbitrary rings, and then distinguish between formal polynomials and polynomial functions. –Henning Makholm 00:02, 20 March 2007 (UTC)
When I came to it the intro was the following paragraphs:

Lindemann and Hiroshi Umemura showed that the roots may also be expressed in terms of Siegel modular functions, generalizations of the theta functions that appear in the theory of elliptic functions. These characterizations of the roots of arbitrary polynomials are generalizations of the methods previously discovered to solve the quintic equation. Equations of degree two can without computer be reduced to square roots. See quadratic equations.

I figured it must have been vandalized, or mistakenly rearranged, but when I checked the history, it had gone through several edits through several different people without being caught. Then the first sensible version I found had the current definition, which seems a bit inadequate.
Should a distinction between addition and subtraction be made? I'm of the philosophy that algebra students should be in the practice of not makiing a distinction between the two, it complicates things far to much, and if far less elegant, to think of subtraction as binary operator. Students start making those basic mistakes as soon as they get out of the habit of thinking of it as such and I think math articles shouldn't reinforce the habit. Brentt 04:15, 20 March 2007 (UTC)

I agree in principle that subtractions should be treated as addition of the opposite, but remember that this article is being read by (I assume) large numbers of seventh and eighth graders. The intro should be reader friendly. Rick Norwood 12:22, 20 March 2007 (UTC)

So let me get this straight, they can't think of subtraction as addition of the additive inverse, but they can understand exponentiation is "just multiplication", not to mention know that having a coefficient that is a non-integer rational number isn't "division". I don't know, seems a bit bss-aackwards. Brentt 17:23, 20 March 2007 (UTC)

I have tried to fix both of the problems you mention. And I will include the information that just addition and multiplication suffice. Rick Norwood 12:39, 21 March 2007 (UTC)

Starting a rewrite

I agree that the article is lumpy and uneven, though I think the writers' intentions were good. To make the article useful, it should begin with a simple but mathematically accurate definition and example, and introduce early on the vocabularly of a polynomial: term, factor, coefficient, exponent, degree, monomial, binomial, number of variables, etc. The important thing is to be accurate but not to confuse the beginning reader. Rick Norwood 12:30, 20 March 2007 (UTC)

I am going to pause here for a little while. I find it is always a mistake to try to do too much in one day. I realize that there is a lot of repetition in the article now which will need to be eliminated later.

By the way, I've worked on this article before, and over time people have rewritten and rewritten and rewritten until it turned out all lumpy. I guess one of the characteristics of a wiki is that you must constantly reinvent wheel.

Why are the equations coming out in different sizes? The code looks exactly the same. Rick Norwood 13:05, 20 March 2007 (UTC)

I can do a rewrite of the "Overview" section, but I will need to know the consensus on a few things first. As I understand it, the polynomial can be expressed in many ways, e.g., (x-1)(x+1)=x2-1. Are we only calling the fully expanded form a polynomial, or is every equivalent expression also a polynomial? Also, I think there should be several illustrative examples of what is and is not a polynomial very early in the article. Tell me what you think.

First, let me suggest you sign your comments with four tildes. The "Overview" section has had a lot of work recently, and I don't think it needs more right now. The article currently states that an expression which can be written in polynomial form is often called a polynomial, and gives several examples of expressions that are polynomials and expressions that are not polynomials. Rick Norwood 22:06, 10 April 2007 (UTC)

I'm an anonymous math enthusiast, but I'll probably get an account soon if I find myself wanting to rewrite more articles. From the history I can see that the overview has been redone many times, but I still don't think it's a good introduction for a student or anyone who wants to quickly reference what a polynomial technically is. I'll consider making some small changes, which you all could revert of course if they are unnecessary.
I agree that the "overview" section it itself is excellent. Unforfortunately it seems not to agree with the lead (which describes the "generic +,-,* expression" definition), or with some of the following sections which start out by saying that any product of polynomials is a polynomial. Immediately, these claims to not seem to be compatible with the explanation that a polynomial is a sum of monomials. This probably does not confuse any of the esteemed participants in this discussion, but it will needlessly confuse some readers who would reasonably expect an encyclopedia to tell them some basic facts about polynomials. I move that we bring the rest of the article into line with Rick Norwood's rewritten introduction. –Henning Makholm 22:25, 10 April 2007 (UTC)
To the anon: Please do sign with four tildes even through you are not logged in. It makes talk pages much easier to follow. –Henning Makholm 22:25, 10 April 2007 (UTC)

I reverted your "corrections" because I am sorry to say they are not correct according to current US usage. First, when we write a to the b equals c, a is called the base, b the exponent, and c the power. Thus 8 is the third power of 2.

Some authors call constant polynomials degree zero, others don't, but I really don't think we need to go into points on which authors disagree in the introduction. Rick Norwood 12:32, 11 April 2007 (UTC)

Equations of different sizes

I hoped that by now someone would have explained how to get all the equations the same size. Help? Rick Norwood 22:10, 10 April 2007 (UTC)

Hi, Rick. I'm not positive I know what you mean. Some of the polynomials are rendered via HTML (with the sup and sub tags) and others are rendered as graphics. In the latter case, the polynomials tend to look larger. To force all math to render graphically (and not just use HTML) you can put a \, inside the math tag. For instance (check out the code for these):
x6 − y5 = 3
versus
x^6-y^5=3\,
Does that help? Cheers, Doctormatt 22:19, 10 April 2007 (UTC)
But please don't do that. We should trust the ability of the wiki software's math renderer to display formulas as best it can according to the user's preferences. If it doesn't do that right, what should be fixed is the software rather than a thousand markup-abusing workarounds. If you want to see all math rendered as graphics, you can request that using the "my preferences" link. –Henning Makholm 22:28, 10 April 2007 (UTC)
How do I fix the software? Quite often, the HTML version looks lousy, and the rendered version looks much better. What should I do in that case? I can set my own preferences, but should I just let it look lousy for those who don't set their preferences to always render? What can I do to help this situation? Thanks. Doctormatt 22:42, 10 April 2007 (UTC)

I'm faced with the same problem. The only solution I can think of is to change "my preferences" and the retype the formulas. Rick Norwood 12:25, 11 April 2007 (UTC) I just looked at the math preferences available, and am not sure which one to use. Any thoughts on that subject? Rick Norwood 12:39, 11 April 2007 (UTC) Until something better comes along, I'm going to use the backslash comma to force display, so that all of the displayed mathematics in a given section is the same size.

Does the change in "my preferences" change how what I type appears to others, or does it only change how everything appears to me? Rick Norwood 12:53, 11 April 2007 (UTC)

This must only change how you see it, since this information is not encoded in the article.
On Wikipedia:Manual_of_Style_(mathematics), it specifically says:
If you want to force an image output for a simple formula, put, for example, a \, (one quarter space in LaTeX) at the end of the formula.
and says nothing about the problems that might be associated with this, that Henning suggests. Perhaps if this really is likely to be a problem (for future compatability) someone should point that out on that talk page, or perhaps on the style page itself? Doctormatt 20:24, 11 April 2007 (UTC)
What continues to baffle me is that formulas that look exactly alike, on the edit frame, display differently on the page. Rick Norwood 21:28, 11 April 2007 (UTC)
Generally consistancy is a good thing. Unfortunately wiki markup lacks the ability to say "display all theese formulas in the same style as they are all meant to be read together", instead you get a horrible mixture with the simpler formulas in one style and the more complex ones in another. It is all very well to say improve the software but sadly that is very unlikely to happen. Plugwash (talk) 02:05, 11 January 2008 (UTC)

Monic polynomial isn't defined

Monic polynomial (referenced from Minimal polynomial) redirects to this page, but this page doesn't define the term. A text search "monic polynomial" or "monic" comes up empty.

Thanks for pointing this out. User:Silly rabbit has fixed the problem. Cheers, Doctormatt 00:50, 13 May 2007 (UTC)

Integer vs. Positive Integer

In the Overview section, it says that

the expression

( 5 + y ) ^ n \qquad\, where n is an integer

is a polynomial (in y) since the exponent is explicitly declared as an integer.

But shouldn't that say "where n is a positive integer"? Because if n were negative, it would no longer be a polynomial (at least from what I learned in school). Foxjwill 01:33, 1 June 2007 (UTC)

    • It should say "non-negative", as constants are the coefficients of x^0 —Preceding unsigned comment added by 65.82.119.254 (talk) 17:14, 27 November 2007 (UTC)

Fundemental Theorem of Algebra

The "Fundemental Theorem of Algebra" states that "any polynomial with real or complex coefficients has a root in the complex plane." This wiki article says that 4 is a monomial. This I agree with. It is also stated that monomials are polynomials. This I agree with as well. Thus 4 is a polynomial. I agree still. The coefficients of "4" are real, so it should follow that "4" has a root in the complex plane. However, 4 has no roots. I do not know whether there is a false definition here, or whether there exists a trivial flaw in the Fundemental Theorem of Algebra. Eat2thepieseye 04:21, 11 June 2007 (UTC)

Allways good to check for exceptional cases. The Fundamental theorem of algebra states that a ploynomial has exactly as many complex roots as its degree, if repeated roots are counted up to their multiplicity, 4 has degree 0 hence zero roots. This article states the weeker result that every (non-constant) polynomial has at least one distinct root so again everything is OK. --Salix alba (talk) 07:18, 11 June 2007 (UTC)

It is common for either of the two claims to be labeled "The Funamental Theorem of Algebra"; one source that gives the "weaker" result is Complex Analysis by Ian Steward and David Tall, Cambridge University Press 1983. (I put "weaker" in quotes, because the other form of the theorem follows from it trivially by polynomial division). –Henning Makholm 21:27, 11 June 2007 (UTC)

Doctormatt's edit

1) This seems a strange place to introduce the vocabularly word "primative" but I won't object. What I do object to is the article "the", since a polynomial has more than one antiderivative. I've changed "the" to "a". 2) While it is true that some polynomials factor into linear factors, and all polynomials do so over the complex numbers, there are polynomials over the reals that factor, but not into linear factors, for example x4 + 2x2 + 1 The point here is that (x2 + 1)(x2 + 1) is still considered a polynomial in factored form. If you want to discuss linear factors, there may be a place to do so, but this isn't it. Rick Norwood 12:39, 14 August 2007 (UTC)

Sounds good to me. Cheers, Doctormatt 16
58, 14 August 2007 (UTC)

Division by expressions containing variables

In the definition of polynomial it currently states, "Note in particular that division by an expression containing a variable is not generally allowed in polynomials." (emphasis added.) This statement would seem to imply that there are exceptions to the general case. I'm not an expert on polynomials by any means, but I think we should remove the word "generally" because it seems to serve only to hedge our bets if this definition turns out to be wrong. In high school math, I think we learned that if there is any division by a variable, then the expression is considered a rational expression, and it is not a polynomial. WilliamJenkins09 22:16, 12 November 2007 (UTC)

I agree and have removed "generally". I can think of two things this might have been intended to hedge: (1) The expression (x4 − 1) / (x2 + 1) (x real) defines a plynomial function, because it is always equal to x2 − 1. (2) An expression such as (x − 5y) / (y2 + 3) is a good and proper polynomial in x, for as long as we hold y constant. Neither of these corner cases really seem to deserve such a confusing hedge in the lead paragraph. –Henning Makholm 23:53, 12 November 2007 (UTC)

I'm not sure why "generally" is confusing, and I've reinstated it in the interest of mathematical correctness. Your examples of exceptions are the reason for the "generally". Rick Norwood 16:39, 13 November 2007 (UTC)

I don't believe it's correct, or that (x^2-1)/(x-1) is a polynomial. This relates to the comment below. — Carl (CBM · talk) 13:29, 14 November 2007 (UTC)

Equality of polynomials, also polynomial functions

The article hedges the issue of whether a polynomial is a particular expression (the correct view, in my mind) or an equivalence class of expressions. I would not say that x^2-2x+3 and (x+1)(x-3) are equal polynomials; they are not equal any more than the words abb − 1 and a are equal words in the free group on two generators. The two polynomials are equivalent in the same way those two words are equivalent. If a polynomial were considered as an equivalence class, it might make sense to say (x^2-1)/(x-1) is a polynomial. But when they are considered as expressions it doesn't make much sense. — Carl (CBM · talk) 13:29, 14 November 2007 (UTC)

On further thought, part of the issue is that the article doesn't distinguish much between polynomials, which are syntactic expressions, and polynomial functions. Equivalence of polynomials can be stated simply as "they produce the same polynomial function". I do agree that (x^2-1)/(x-1) induces a polynomial function when x is not 1. — Carl (CBM · talk) 13:41, 14 November 2007 (UTC)

To try to make a distinction between equal and equivalent at this level is not a good idea, it is needlessly confusing to the beginner and unnecessary for the mathematician. It would be like going through Wikipedia and changing references to the "number" 2 to read "numeral" 2 -- technically correct, but against standard practice. Similarly, everyone who works with free groups calls the two elements you give above in the free group on two generators "equal" in everyday practice, unless the distinction between equal and equivalent becomes important, which it rarely does. See, for example, Magnus, Karrass, & Solitar, Combinatorial Group Theory, page 19. Rick Norwood 13:55, 14 November 2007 (UTC)

But the difference between polynomial expressions and polynomial functions is important in even elementary mathematics. It's a common source of confusion for students in calculus and abstract algebra. And it's much less esoteric than the distinction between the number 2 and the numeral 2.
I agree this article should be accessible to a very wide audience, but I think that can be accomplished while being somewhat more precise than the current article is. That is, I don't think we have to ignore the distinction. I think we can mention it in passing but not dwell on it. Then people who are not interested can move on quickly to stuff they are interested in.
I'd like to know what other people think. — Carl (CBM · talk) 14:09, 14 November 2007 (UTC)

copyediting, lede

After reading through the article in detail, I think it does a great job of including the right information. But I think the first few sections are somewhat fragmented; they come across as a collection of juxtaposed independent sentences rather than an article. I want to try my hand at copyediting the prose while keeping the level of the material just as accessible as it is now.

Also, the lede section of an article is meant to be a brief overview of the entire content of the article. The lede here doesn't meet that goal. It's particularly important for the lede to be accessible, but also important for it to be compelling. — Carl (CBM · talk) 14:22, 14 November 2007 (UTC)