Talk:Polynomial/Archive 1

Quadratic formula
At the high school level quadratic equations are useful in displaying the teacher's facility in proving the quadratic formula, by completing the square. No thanks necessary.
 * Early usage: For LarrySanger

Questions (I'd rather not make changes to the page since it's out of my area of expertise):
 * 1) Can I have more examples please? See below.
 * 2) What are polynomials good for? Describing some kinds of relations between variables. Like much of math, we present definitions and techniques and leave the applications to others.
 * 3) Maybe an example of a high school textbook problem involving polynomials would help orient me...just a suggestion, feel totally free to ignore it.

Further examples
Further examples of polynomials (some are monomials which form a special case with only one term):
 * Area of a square = side2
 * Volume of a cube = side3.
 * Area of a square with its lower left corner at the point (x,x) and its upper right corner at (y,y) = (y-x)2 = y2-2xy+x2.
 * This is a polynomial in the two variables, x and y.

AC Method
If somebody wants to integrate my writeup on E2 to here, feel free. The AC Method may be of particular interest. This is primarily just telling how to factor polynomials so there might be a better place (i.e. factoring) to put it. For simplicity, I'll post a partially wikified version here. If you think it's useful, integrate it. Else, just remove it: http://everything2.org/?node_id=895118 (Note: could contain some errors.)

anxn + an-1xn-1 + an-2xn-2. . . a1x + a0

The degree of a polynomial is the highest total of powers of variables (x, y, etc.) of a single term, so in the polynomial 2xy<SUP>2</SUP> + x<SUP>2</SUP> the degree is three (in the first term, x has a power of one). The standard form of a polynomial is when you write it with the degrees descending (x<SUP>2</SUP> + x + 3, not x + x<SUP>2</SUP> + 3)

To factor a polynomial (If you already know how to then skip down to the AC method. You'll like it. A lot.) you first factor out the common factor, if there is one, using the distributive property:

<B>Ex 1)</B> 2x2 + 4x = 2x(x + 2) <B>Ex 2)</B> 2x<SUP>2</SUP> + 6x + 8 = 2(x<SUP>2</SUP> + 3x + 4)

With a binomial (two terms, as in <B>Ex 1</B>) that's all. If you have a trinomial (three terms, as in <B>Ex 2</B>) you're just getting started.

You usually have to find two binomials (B<SUB>1</SUB> and B<SUB>2</SUB>) whose first terms multiply to the first term of your trinomial, last terms multiply to the last term of the trinomial, and B<SUB>1</SUB>'s first term times B<SUB>2</SUB>'s last term plus vise versa equals the middle term (FOIL users: Inside + Outside=Middle)

<B>Ex 3)</B> x<SUP>2</SUP> + 3x + 2 = (x + 1)(x + 2)

If the first term of your trinomial has a coefficient (a) of 1--as shown above--then the first terms of the binomials are x. Otherwise, you have to play around searching for the proper factors to get it right. That's where the following method comes in: <P align="center"><B>The AC Method</B></P>

First factor out the common factor. Always, always, always do this.
 * Now you have ax<SUP>2</SUP> + bx + c a isn't 1.
 * Change it to x<SUP>2</SUP> + bx + ac. (If you're stuck wondering how the hell to move the <I>a</I> all the way over to the <I>c</I>, don't bother. Just do it.)
 * Factor x<SUP>2</SUP> + bx + ac into your (presumably) two binomials. Then stick <I>a</I> back into the first terms of both of them, factor out the common factor and toss it out. You're done.

<B>Ex 4)</B> 6x<SUP>2</SUP> + 2x-4
 * 2(3x<SUP>2</SUP> + x-2) (Factor out common factor)
 * 2(x<SUP>2</SUP> + x-6) (move <I>a</I> to third term)
 * 2(x + 3)(x-2) (factor)
 * 2(3x + 3)(3x-2) (put <I>a</I> back into first terms)
 * 2(x + 1)(3x-2) (factor out and delete common factor)
 * If you're planning on using the AC Method a lot you may want to work on your factoring large numbers because <I>ac</I> is often rather large.

Now, I know you're thinking, "What if I have a four-term (or more) polynomial?" Easy: Take a few terms, and slap parenthesis around them (Hint, put together terms that have common factors or that look like they'll factor easily.)

<B>Ex 5)</B> 2x<SUP>3</SUP> - 3x<SUP>2</SUP> + 4x - 6
 * (2x<SUP>3</SUP> - 3x<SUP>2</SUP>) + (4x - 6)
 * x<SUP>2</SUP>(2x - 3) + 2(2x - 3)
 * (x<SUP>2</SUP> + 2)(2x - 3)

That last example (first and last steps anyway) was taken from <I>College Algebra</I> by Michael Sullivan because I was having a heck of a time making up a good example. (I'm always coming up with prime polynomials in my example and having to modify them so I can factor them. I wish my math teacher had let me do that in my homework.)

Now you need to do some heavy memorising. These are special polynomials and how to factor them. Knowing how to recognise them will help you enormously, both in multiplication and factoring:

Difference of Squares: x<SUP>2</SUP> - a<SUP>2</SUP> = (x - a)(x + a) (<B>Ex 6)</B> x<SUP>2</SUP> - 144 = (x + 12)(x - 12))
 * Perfect Squares: x<SUP>2</SUP> &#177; 2ax + a<SUP>2</SUP> = (x &#177; a)<SUP>2</SUP>
 * Unnamed, but bears remembering: x<SUP>2</SUP> + (a + b)x + ab = (x + a)(x + b)
 * Unnamed, but bears remembering: acx<SUP>2</SUP> + (ad + bc)x + bd = (ax + b)(cx + d)
 * Perfect Cubes: x<SUP>3</SUP> + 3ax<SUP>2</SUP> + 3a<SUP>2</SUP>x + a<SUP>3</SUP> = (a + x)<SUP>3</SUP>, x<SUP>3</SUP> - 3ax<SUP>2</SUP> + 3a<SUP>2</SUP>x - a<SUP>3</SUP> = (a - x)<SUP>3</SUP>
 * Sum of Two Cubes: x<SUP>3</SUP> + a<SUP>3</SUP> = (x + a)(x<SUP>2</SUP> - ax + a<SUP>2</SUP>)
 * Difference of Two Cubes: x<SUP>3</SUP> - a<SUP>3</SUP> = (x - a)(x<SUP>2</SUP> + ax +a<SUP>2</SUP>)

Take the coefficients of (x + y)<SUP>n</SUP> and look at the <I>n</I>th row of Pascal's Triangle (the "1" at the top is 0th). Cute <EM>and</EM> useful.

Sorry for the flood. :-) --- If this flood might be useful to someone, maybe it belongs on related corrolary pages. stevertigo --- Technical point: I've always seen a polynomial defined as an expression, not an equation or a function, ie anx^n + ... + a0. The term "polynomial" is later loosely applied to graphs, functions and equations with a polynomial. -- user:Tarquin --- Just wanted to draw everyone's attention to the fact that an anonymous user just changed "In algebra" to "In calculus". with so many mathematicians at Wikipedia, I find it difficult to believe that such an elementary mistake exists in a basic article, and it sounds like something a semi-educated person might think is true.  Personally, I haven't the foggiest notion of what calculus is, much less if... calculators? (calculites? calculians?) use polynomials or not. Tokerboy 03:01 Nov 22, 2002 (UTC)


 * See calculus :-) -- Tarquin

Algebra is a subject. Calculus, on the other hand, is something of a hodge-podge --- a collection of subjects that the curriculum brings together. Algebra goes far beyond those things that most students see, and is a subject to which careers of some researchers are devoted. The topics that go far beyond calculus, on the other hand, are not called "calculus", but go by other names, such as "analysis" and "topology". Therefore, it makes sense to say "in algebra", but not as much to say "in calculus". Polynomials of course appear in calculus, as do many things from algebra. -- Mike Hardy

The reason I separated calculus and algebra is that in algebra, one has to distinguish between polynomials and polynomial functions, while in calculus one doesn't. This point is now lost, in fact the first sentence seems to suggest that the two concepts are the same, which they are only in sloppy calculus usage. AxelBoldt 23:41 Nov 30, 2002 (UTC)

---

I think this article would be improved if some knowledgeable person would add a few sentences about the Fuchsian Function solution to the paragraph which discusses roots of nth order polynomials. They are hinted at with the existing phrase "degree 5 eluded researchers for a long time", which suggests that a solution was eventually found, but this solution is not mentioned in the article. A new article on Fuchsian Funtions would also be welcome. kielhorn@portland.quik.com Dec 22, 2002

Does anyone know anything about "polynomial arithmetic modulo 2" - you know, the mathematics used for cyclic redundancy checks? Because I don't, and it's not explained in the CRC article, either. -- Tim Starling

Removing this:


 * Polynomials are important because they are the simplest functions: their definition involves only addition and multiplication (since the powers are just shorthands for repeated multiplications).

Polynomials are surely not the "simplest"; surely f(x) = 0 is "simpler". In addition, something being simple does not imply that it is important. This sentence contributes nothing of value. -Ryguasu 21:11, 13 Sep 2003 (UTC)


 * f(x) = 0 is a polynomial... - guest

Degrees past the 4th
So far, the polynomial page has special names for degrees up to the 4th. How about past 4th?? Do any of these make sense?? Degree Names from 1 to 12
 * 1) Linear
 * 2) Quadratic
 * 3) Cubic
 * 4) Quartic
 * 5) Quintic
 * 6) Sextic
 * 7) Septic
 * 8) Octic
 * 9) Nonic
 * 10) Decic
 * 11) Unidecic
 * 12) Duodecic

66.32.148.219 00:54, 10 Apr 2004 (UTC)


 * Possibly, but people don't use "special" names for high-degree polynomials. Compare with n-gon. People just say a nth degree polynomial, or the polynomial has degree n. Dysprosia 00:57, 10 Apr 2004 (UTC)


 * The furthest I've heard is "quintic". -- Tarquin 08:08, 10 Apr 2004 (UTC)


 * Sextic, yes. I'd be careful from there on. It may be octavic, for example, for degree 8 sometimes. Septimic, too. Charles Matthews 08:24, 10 Apr 2004 (UTC)

Several critics:
 * is this discussion useful?
 * isn't this an issue that is not strictly related to polynomials and thus just a link should be provided here to some page in linguistics or numbering theory where this is discussed?
 * sextic might be misinterpreted and even caucht by parental filters...
 * it would be more interesting to have an explicit formula for the roots of a general quintic polynomial, and only then go on to higher degrees....  &mdash; MFH: Talk 14:18, 27 May 2005 (UTC)
 * I was under the impression that the quintic was notable as the first polynomial for which such a formulae doesn't exist (at least not expressed in terms of well known functions). Plugwash 00:55, 9 March 2006 (UTC)

Complexity
This paragraph seems a bit confused - is it talking about computational complexity, or bounding general polynomials in magnitude by their leading term?

Charles Matthews 08:24, 14 Jul 2004 (UTC)


 * I moved the Complexity paragraph to big o notation as an example. MathMartin 16:05, 11 Nov 2004 (UTC)

Restructuring
I think the article is in horrible shape. I have rewritten the definition and restructured the existing material. Some key points of the article should be
 * 1) history (finding roots, galois theory)
 * 2) numerical analysis
 * 3) abstract algebra

MathMartin 21:47, 16 Aug 2004 (UTC)


 * Nice work! :) I agree that overview and history should come first: they are the parts the layperson can understand.-- Tarquin 21:56, 16 Aug 2004 (UTC)

Definiton and history
I think the definition of a (mathematical) term should always be the first subsection of its entry. There must also be an Analysis section under mathematics to which the polynomials entry should be moved. Who's responsible for that?


 * It works fine as it is. There may be a way of casually introducing what a polynomial is, before the formal definition sections, however. Dysprosia 01:56, 24 Oct 2004 (UTC)

Scary definition
I do not like the latest edits on the definition. Polynomials are a basic topic which should to accessible to a wide range of people. But now the definition will scare off even undergraduate math students. We should have a simple definition which covers the most common cases and terms and then later in the article we can always add the scary stuff for the fearless.MathMartin 16:12, 11 Nov 2004 (UTC)

Yes, the discussion doesn't cut it. Charles Matthews 22:33, 11 Nov 2004 (UTC)

I'll third that. Paul August 22:47, Nov 11, 2004 (UTC)

I reverted the definition to the more simple one I wrote some time ago. Perhaps someone else can integrate the more abstract definition below into the article. MathMartin 13:43, 17 Nov 2004 (UTC)

I admit, the definition below is quite messy (even incomprehensible since I forgot to say what some of the letters used denote) but there should be some kind of general and precise definition of what a polynomial is. Do you want me to have another go in a new paragraph (eg Polynomial, general definition) or try again to integrate it in the existing Defn paragrah? Ncik

Sure, but please don't remove the accessible definition at the top. Perhaps you can integrate your definition into the Abstract algebra paragraph or create a new paragraph Generalization. MathMartin 10:32, 31 Jan 2005 (UTC)

Let us first note that in most cases the term polynomial refers to a term of the following form:


 * $$\sum_{i=1}^n{a_i x^i},\qquad n\in\mathbf{N},a_i\in\mathbf{Q}.$$

However, this use is, although common, somewhat unprecise since what is actually meant is an univariate polynomial over Q according to the general definition:

Let r, s and t be elements of N, x1,...,xr be variables, F a field and


 * $$M=\{\prod_{j=0}^r x_j^{k_j}: k_j\in\mathbf{N}\land 0\le k_j\le s\}.$$

Furthermore let the finitely many elements of M be denoted by y1,...,yt. Then an r-variate polynomial over F is a term of the form


 * $$\sum_{i=1}^t{a_i y_i},\qquad a_i\in\it{F}.$$

The ai are called coefficients. The coefficient of


 * $$\prod_{j=0}^r x_j^0$$

is called constant coefficient. Polynomials with only one, two or three non-zero coefficients are called monomials, binomials and trinomials, respectively.

The term polynomial can also refer to a function p: M->N, x->p(x), where p(x) is a polynomial as defined above. Such a function may also be called a polynomial function.

The leading coefficient of an univariate polynomial is the coefficient ak which doesn't equal 0 and also has ai=0 for all i > k. We say that a univariate polynomial has degree (or order) k if ak is its leading coefficient, and we say that it is monic or normed if its leading coefficient is 1.

Univariate polynomials of
 * degree 0 are called constant functions,
 * degree 1 are called linear functions,
 * degree 2 are called quadratic functions,
 * degree 3 are called cubic functions,
 * degree 4 are called quartic functions and
 * degree 5 are called quintic functions.

rigurous definition
The question is not whether a definition is "scary" or not, but wether it is a mathematically valid definition or not.

I think we should maintain that "in mathematics" (*sigh*) a polynomial is not a polynomial function.

If we don't at least agree on this, WP becomes useless as a mathematical source of reference.

I say well "agree" and not "write", I mean there can be lots of handwaving and blabla, but (in articles on mathematics) the section entitled "Definition", even if it's at the very end of the article, should really be strictly reserved to a true, mathematical definition on which all textbooks on the subject agree (not only those for primary schools).

I mean, the whole definition, if it's not the true definition, is not worth more than saying "a polynomial is something like x²+5x+3, or 3.9x^7 - 0.01, or any similar expression".

A polynomial is, and will ever remain, a map from N (or some Cartesian power thereof), into a ring (at least), with canonical structure of module and convolution style multiplication. (Or does anybody prefer a terminology involving mysterious abstract "symbols" X which under some obscure conditions can be the same than the symbol 'Y' or even 't', and under some other conditions are different from 'Y' ?)

Once again, I don't mean to explain it like this in the first section, but please, at least allow to mention that there should be a distinction of 'polynomial' from 'polynomial function', even if, by abuse of language, and because on R and C they can be identified, the remainder of the article uses "polynomial" instead of "polynomial function". &mdash; MFH: Talk 23:12, 24 May 2005 (UTC)


 * The purely algebraic definition of a polynomial and the discussion about polynomial vs polymomial function is further below in the article, also in polynomial ring.


 * I understand your point. For myself, I don't like rigor in Wikipedia. :) For rigor one could go to PlanetMath, MathWorld, or read a book. :) Oleg Alexandrov 23:29, 24 May 2005 (UTC)


 * More exactly, the rigurous definition is in Polynomial. Oleg Alexandrov 18:00, 25 May 2005 (UTC)

Perhaps we should change the first sentence to say "In mathematical analysis …" and perhaps refer to the more general notion defined below? Would this help? Paul August &#9742; 20:29, May 25, 2005 (UTC)


 * You can give it a try. :) Oleg Alexandrov 03:07, 27 May 2005 (UTC)

Changed content
Under section Graphs --> Number of x-intercepts,

Earlier erroneous content: For example, a degree 4 polynomial function can have 0, 1, 2, 3 or 4 x-intercepts whereas a degree 5 polynomial function can have 1, 2, 3, 4 or 5 x-intercepts.

Have changed it to: For example, a degree 4 polynomial function can have 0, 2 or 4 x-intercepts whereas a degree 5 polynomial function can have 1, 3 or 5 x-intercepts.

--Wowbagger


 * you are correct --MarSch 15:15, 1 November 2005 (UTC)
 * This is not true. Consider the polynomial function y=x^4. This has only 1 x-intercept. Even if you insist it has multiplicity 4, and thus has 4 x intercepts, this is still only true for polynomials with real coefficients. Consider y=x^3(x-sqrt(-1)) This has degree 4 and 1 x-intercept with only multiplicity 3. This should be reverted to say that a polynomial function has x-intercepts (or roots) which number less than or equal to the degree of the polynomial function.Eat2thepieseye 04:31, 11 June 2007 (UTC)

Reworded intro
I would argue that the introduction of this article was too biased towards numerical analysis, with two paragraphs devoted to that, and there was no mention of abstract polynomials, which are no longer smooth polynomial functions. I think this issue came up earlier, raised by MFH. Anyway, I cut one of the two numerical analysis paragraphs and replaced it with a blurb on abstract algebra. Comments welcome. Oleg Alexandrov (talk) 20:16, 19 December 2005 (UTC)

Form and function
polynomial function redirects to polynomial. But Rational expression redirects to rational function. I don't care which way the redirects go, but it should be consistent. My personal preference is for the object to be the title of the article and the function to be a major subtopic, but I can go either way. I'm placing this comment in the talk pages of both polynomial and rational function in hopes of finding a consensus. Rick Norwood 16:15, 24 December 2005 (UTC)


 * Consistency when it comes to redirects, heh?


 * I like things the way they are. The most widespread of "polynomial" and "polynomial function" is "polynomial". The most widespread of "rational function" and "rational expression" is "rational function", even if it is not always a function.


 * I think all this can be easily solved by inserting in rational function, at the section about abstract algebra, the remark that in the abstract algebra context a rational function is not really a function but rather a rational expression. Any big changes for the sake of some hypothetical need for consistency are, in my opinion, not necessary. Oleg Alexandrov (talk) 23:43, 24 December 2005 (UTC)

questions
"...Laguerre's method which employs complex arithmetic and can locate all complex roots."

This can't be quite right. Approximate all complex roots?

Also, is there a difference between "total degree" and "degree"?

Answer: The expression xy2 defines several polynomials: x&rarr;xy2 of degree 1, y&rarr;xy2 of degree 2, and (x,y)&rarr;xy2 of degree 3. Bo Jacoby 10:24, 28 February 2006 (UTC)


 * I assume that last is the "total degree". Thanks. Rick Norwood 14:12, 28 February 2006 (UTC)

Using polynomials for extending the concept of number
I am not happy with the new section ==Using polynomials for extending the concept of number==. It appears to me like some observation of an amateour mathematician, rather than a serious topic about polynomials belonging in this article. Oleg Alexandrov (talk) 17:57, 27 February 2006 (UTC)


 * I moved it further down the article rather than delete it entirely -- but if somebody did delete it entirely I would not object. It is a modestly interesting idea, but badly written in places. Rick Norwood 20:52, 27 February 2006 (UTC)

I could't find the construction of algebraic numbers from natural numbers anywhere else in wikipedia, so I gave it here. The use of polynomials for this purpose is elementary and central, and the idea is standard mathematics and not just 'modestly interesting'. The connection to the theory of ideals needs to be explained further. Tell me the places where you find it badly written so that it can be improved. Spell 'amateur'. :-) Bo Jacoby 09:59, 28 February 2006 (UTC)

Oleg, I took the liberty to change your heading for this section in the discussion page from expressing your emotion and into the subject matter. Bo Jacoby 10:17, 28 February 2006 (UTC)

The construction of the algebraic numbers belongs in algebraic numbers. One "badly written" sentence I noticed was "From this description, addition and multiplication of natural numbers is defined, and the elementary rules of arithmetic are proved." Should be "addition and multiplication...are defined..." There were several others. I don't mean to be too critical -- I did find the section interesting. But it needs to be shorter and more focused, and maybe somewhere else. Number systems might be a good place to start with the Peano axioms, construct negative numbers, pass to the quotient field of the integers, then to polynomials over a field, then to algebraic numbers, and then use Dedikind cuts to get the real numbers and extension fields to get the complex. Then all that is needed here is a reference. Rick Norwood 14:21, 28 February 2006 (UTC)


 * I deleted that section. It was way too long, and not central to the concept of polynomial, rather, a distraction. Such "insightful extrapolations" belong on a blog or something, not in an encyclopedic article. Let us keep focused, and if possible, provide references. Oleg Alexandrov (talk) 18:57, 28 February 2006 (UTC)

There is no link to algebraic numbers in the polynomial article. In the sentence "From this description, addition and multiplication of natural numbers is defined", the word "defined" refers to "addition" and "multiplication" (singularis), and not to "natural numbers" (pluralis). I'm glad, Rick, that you found the section interesting. Other readers might find it interesting too. The crux of the peano axioms is that every natural number has a successor. However, I don't approve of Guiseppe Peano's habit of letting the first natural number be 0 rather that 1 (if he was really the one who did it?). Counting to three is saying: "one, two three" and not: "zero, one, two". Thanks, Rick, for the link to number system. The point is that the number extensions to integers, fractions, square roots, and complex, algebraic numbers are defined by polynomial (algebraic) equations. This is a central application of polynomials, which I think belongs in an encyclopedic article on polynomials. I don't mind if the explanations are replaced by references. Please provide references, Oleg, to make your net contribution to this matter become positive rather that negative. Bo Jacoby 08:42, 1 March 2006 (UTC)


 * Bo, I think you have misunderstood Oleg's point about providing references. The point that he was trying to make was, I believe, that your new section looked, on the face of it, like original research, which would disqualify it from appearing anywhere in Wikipedia. To demonstrate that it is not original research, the onus is on you to provide references showing that this approach can be found somewhere in published literature (papers or textbooks). FWIW, I also think the section was confusing, and I agree with its removal. Gandalf61 13:26, 1 March 2006 (UTC)

Thank you. The idea is the standard one of defining the difference X=a&minus;b by the polynomial equation X+b=a. So manipulation of differences and negative numbers reduce to manipulation with equations of polynomials having natural number coefficients. The same method applies to fractions and to algebraic numbers in general. A reference is number system. I am amazed that you were confused. Please tell me why. Bo Jacoby 14:08, 1 March 2006 (UTC)


 * Well, one confusing aspect was that your basic object is a polynomial over the natural numbers. The natural numbers aren't a ring, so your "polynomial with natural number coefficients" does not fit the usual algebraic definition of polynomial - your "polynomials" aren't even a group under addition. Another confusing aspect is that you exclude 0 from your definition of natural numbers, so your set of "polynomials" doesn't initially even have an additive identity. And I don't understand why you think number system supports your case - that article follows a much more standard exposition than yours, and does not even mention polynomials until after the rational numbers have been defined. Gandalf61 16:01, 1 March 2006 (UTC)


 * Gandalf got my point right about references. Bo Jacoby, no offence, but we already intersected at complex number, root-finding algorithm and now here (not counting the discussion at talk: formal power series). I am not happy with you adding your own insights and observations to articles. That borders on original research, and your text tends to be poorly written. When you want to add something to an article, say about polynomial, please read things in a book (say abstract algebra book), then add to the article, and append the book as a reference at the end. Otherwise, please follow Linas, and make your own web page (see http://www.linas.org/) where you can publish your observations. Sorry for being harsh, but this had to be said. Oleg Alexandrov (talk) 03:24, 2 March 2006 (UTC)

To Gandalf: Neither the set of natural numbers, nor the set of polynomials over natural numbers, are rings. The set of integers, and the set of polynomials over integers, are rings. You can define polynomial equations over natural numbers without using (negative) integers, but you cannot define (negative) integers without using polynomial equations over natural numbers, such as X+b=a. So the path of logic has to be: natural numbers &rarr; polynomials over natural numbers &rarr; integers &rarr; polynomials over integers. (That this path is not always followed in number system is perhaps due to history). The discussion on whether zero is considered a natural number is not settled. Most people count "one two three" rather that "zero one two". The additive identity is a formal solution to the polynomial equation X+1=1. I don't have a case to support; I just included this very elementary and very important application of polynomials in the polynomial article. To Oleg: It makes me sad that you are not happy. Wikipedia should be joy and fun. I appreciate that you do a great job for no pay except fun and appreciation, so stop doing it if you don't enjoy it any longer. Our collaboration on complex number and root-finding algorithm resulted in substantial improvements which I found satisfying and I'm sad if you didn't. Thanks for the link to www.linas.org. Bo Jacoby 09:01, 2 March 2006 (UTC)


 * Bo: (1) My point was that the standard definition of polynomial is defined over a ring, so your set of "polynomials with natural number coefficients" uses a non-standard definition of polynomial, which is confusing. (2) It is possible to define negative numbers without using polynomials. (3) Your path is not the only logical path. (4) In colloquial English, "I am not happy with ..." means "I do not agree with ..." - Oleg was not commenting on his emotional state ! Gandalf61 13:48, 2 March 2006 (UTC)

(1) The polynomials over natural numbers constitute a subset of the ring of polynomials over integers. Only an educated mathematician may get confused. You may also say that 1,2,3,4... are integers, although not every integer occur in the sequence. (2) No, you can not define negative numbers without considering the problem X+b=a. You do not need to hear the word 'polynomial', but the concept is unavoidable. It may be disguised as a riddle like the one quoted in Diophantus. (3) If you change the path, you do break the logic. The later concepts depend on the earlier concepts in the path. (4) That's comforting. However, the habit of quickly removing whatever one dislikes discloses a somewhat tense and impatient emotional state, I think. When Oleg says that he is sorry for being harsh I believe that it is true. Bo Jacoby 15:23, 2 March 2006 (UTC)


 * You don't need polynomials to define integers; the usual construction is done with $$\mathbb N\times \mathbb N$$ quotiented by some equivalence relation. But that is not the point. The point is:


 * 1) Please provide references for your construction. Please do. Wikipedia does not allow original research.
 * 2) Your construction does not belong to this article. It belongs to an article about construction of numbers or something.
 * Oleg Alexandrov (talk) 16:37, 2 March 2006 (UTC)

The usual construction of integers (reference ?) is that (a,b) is equivalent to (c,d) when the equation X+b=a implies the equation X+d=c, which is iff a+d=b+c (=X+b+d). There is isomorphism between the set of pairs (a,b), and the set of polynomial equations of the form X+b=a. That is not original research but merely explaining the usual construction. The polynomial equation representation motivates the equivalence relation, while the definition a+d=b+c is artificial for the reader. So far the article contains no motivation for the definitions, which is a pity as polynomials are useful for theoretical and practical mathematics. So why does an application of polynomials not belong to this article ? Bo Jacoby 15:00, 6 March 2006 (UTC)


 * This application of polynomials does not belong here because there are literally thousands of applications of polynomials, of which this one is on the esoteric side. (Please add comments at the bottom of the page.  That's where people look for them.  If what you want to say does not fit the topic currently at the bottom of the page, click on the plus sign after "edit this page" and title the new section something like "Reply to Oleg". Rick Norwood 15:08, 6 March 2006 (UTC)

It does not make sense to me to even involve polynomials in the construction of integers. Yes, (a,b) is equivalent to (c,d) when a+d=b+c. There is no need to complicate matters by saying that this is tied to the solution of a polynomial equation. If you do that, you need to construct the polynomials first, which are finite sequences of integers, define their operations, define the concept of root of a polynomial, etc. That would complicate the construction of integers by an order of magnitute. Oleg Alexandrov (talk) 17:10, 6 March 2006 (UTC)

Issues
The definition says: "In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, and multiplication". As constants may be negative, subtraction is actually not needed. Only addition and multiplication is (are?) needed.

The claim: "More powerful methods for solving several polynomial equations in several unknowns are given in linear algebra" does not seem to be true. The article on linear algebra does not contain the promised methods. Do anyone intend to write it? Am I allowed to give it a try ?

The section does not contain examples of polynomials but merely references and should be renamed and moved far away downwards.

The small section "Notes" needs a better name and a better place, perhaps nowhere?

The section "Evaluation of polynomials" fails to define "evaluation". The statement: "Several different algorithms have been developed for this problem. Which algorithm is used for a given polynomial depends on the form of the polynomial and the chosen x" is not helpful and hardly even true. The reader is happier without it.

The sections "Roots" and "Finding roots" should be merged.

Bo Jacoby 12:37, 1 March 2006 (UTC)


 * A note on plurals: "Monday and Tuesday are days of the week." Not "Monday and Tuesday is days of the week."


 * On the definition of polynomials: It is possible to leave out subtraction, but there is no harm in putting it in, and the stuff about the ToC should be something a layperson can understand.


 * The more powerful method is Gaussian elimination, of course. If Linear algebra doesn't link there, it should.


 * I agree about moving "More advanced examples of polynomials" further down in the article. The rest of your comments I also agree with, and hope you can help improve the article. Rick Norwood 21:10, 1 March 2006 (UTC)


 * OK. I did some editing. Please comment. Bo Jacoby 12:17, 6 March 2006 (UTC)

Is it "addition and multiplication is defined by induction" or "addition and multiplication are defined by induction" ? When expanding the first version by the distributive law, (a+b)c=ac+bc, you get: "addition is defined by induction and multiplication is defined by induction". So: 'is'. Your example "Monday and Tuesday are days of the week" cannot be expanded into "Monday are days of the week and Tuesday are days of the week"  because "days" is (are?) plural. My mothers language was not English, so I am in doubt. Don't hesitate to improve on my English.


 * Verbs do not distribute over "and". For example.  A cat is my pet.  A dog is my pet.  A cat and a dog are my pets. Rick Norwood 20:40, 2 March 2006 (UTC)

Gaussian elimination is limited to the case of linear equations. However the method basicly also applies to the nonlinear case of several polynomial equations with several unknowns. See Buchberger's algorithm and Gröbner basis. Let 0=f(x,y)=g(x,y) be two polynomial equations with two unknowns, x and y. Let A(x,y) be an arbitrary polynomial of x and y. Then the equations 0=f(x,y)+g(x,y)=A(x,y)&middot;f(x,y) follow. So the set (f,g) of polynomials, X(x,y), for which the polynomial equation 0=X(x,y) follow from the original system of equations 0=f(x,y)=g(x,y), is an ideal (ring theory) in the ring Z[x][y] of polynomials of y over the ring of polynomials of x. See Greatest_common_divisor. If this ideal, (f,g), contains an element that is of degree zero in y, then y is said to be eliminated and the resulting equation in x alone can be solved numerically by the Durand-Kerner method. The elimination process is: Assume that the degree of f is greater than or equal to the degree of g. (Otherwise switch names). Multiply f and g with monomials so that they have the same leading term, and subtract g from f. Now the degree of f has diminished. Continue until the degree is zero. This algorithm is similar to Euclids algorithm for finding the greatest common divisor between two natural numbers.


 * Which doesn't change the fact that more advanced methods for solving systems of linear equations are taught in linear algebra. Rick Norwood 20:40, 2 March 2006 (UTC)

Note my edit on number system. Bo Jacoby 10:44, 2 March 2006 (UTC)

Note to Bo Jacoby
If you put your comments at the bottom of the page, they are more likely to be read. That is where people naturally go to look for the newest comments. Rick Norwood 14:11, 6 March 2006 (UTC)
 * Thanks. I'll do that, except for answers, which naturally belong just after the question, I think. Bo Jacoby 15:09, 6 March 2006 (UTC)

Roots
The section on roots makes no distinction between exact solutions and approximate solutions. I'm working on the problem. In the process, I discovered that algebraic equation also needs a lot of work -- just in case anyone is looking for an important article that cries out for a careful rewrite. Rick Norwood 14:37, 6 March 2006 (UTC)
 * I'm not sure that the distinction needs to be made right here. The Durand-Kerner method seems to provide only approximate solutions, but actually it provides convergent sequences which define exact solutions. So we need not talk about approximations here. Bo Jacoby 15:13, 6 March 2006 (UTC)
 * I disagree with your removing my edit. It was discussed above in the discussion page. Discuss first and edit later please Bo Jacoby 15:19, 6 March 2006 (UTC)


 * The distinction between exact roots and approximate roots is an important one. Also, in the general article on polynomials, there is no need to repeat technical details found in referenced articles.  The topic of polynomials is a huge one -- it could fill an entire library -- and this article cannot cover everything. Rick Norwood 15:45, 6 March 2006 (UTC)


 * There is but historical reasons to give degrees 2,3 and 4 special treatments. Degree one must have special treatment, division, but all higher degrees are treated by durand-kerner. Solutions in terms of radicals is a dead end. The polynomials define the roots exactly. Computation of radicals is not easier than computation of roots in general. (That is, the equation xn=a is not easier to solve than f(x)=0 where f is a polynomial of degree n). Therefore the distinction between exact roots and approximate roots is not at all important in this context. I agree that the referenced articles should not be repeated, and I did not repeat. You state that solution of equations is difficult. It is no help to the reader to know that it is difficult to you. Don't write articles while you find the subject matter difficult. To me it is not difficult, it is just hard work. I gave examples of how to solve equations, but you deleted it instead of studying it to see that it is not difficult. Your advice to linearize is bad advice in general. You omit the reference to gaussian elimination. The descartes law of signs is of only historical interest. Bo Jacoby 08:11, 7 March 2006 (UTC)

day 1 of discussion
Oleg: It does not make sense to me to even involve polynomials in the construction of integers.

Bo: The elementary explanation of subtraction is that a&minus;b solves the polynomial equation X+b=a.

Oleg: (a,b) is equivalent to (c,d) when a+d=b+c. There is no need to complicate matters by saying that this is tied to the solution of a polynomial equation.

Bo: That X+b=a and X+c=d have the same solution, motivates the otherwise ad hoc condition a+d=b+c.

Oleg: If you do that, you need to construct the polynomials first, which are finite sequences of integers,

Bo: Polynomials are defined recursively: "If A,B are polynomials, the so is A+B and AB. 1 is a polynomial, and X is a polynomial".

Oleg: define their operations,

Bo: No new operations, just addition and multiplication and the rules of associativity, commutativity, distributivity and cancellation.

Oleg: define the concept of root of a polynomial,

Bo: only the concept of a solution to an equation.

Oleg: etc.

Bo: no etc, the job is done.

Oleg: That would complicate the construction of integers by an order of magnitude. Oleg Alexandrov (talk) 17:10, 6 March 2006 (UTC)

Bo: Understanding integers is not needed for understanding polynomials. Fewer prerequisites means more readers. Bo Jacoby 09:24, 7 March 2006 (UTC)


 * What has this to do with anything? If we want to read previous talk, we can do so. We don't need a summary. Dysprosia 09:34, 7 March 2006 (UTC)
 * As Rick requested me to make my answers at the end of the page, I also repeated Oleg's statements here. Do you have an opinion on the discussed matter, Dysprosia? Should my edit be included or excluded ? Bo Jacoby 09:54, 7 March 2006 (UTC)


 * Rick probably meant to place any new comments you have to have at the end of a page. If someone makes a comment, you usually place your reply below their answer, as you have done there. Dysprosia 10:11, 7 March 2006 (UTC)


 * Actually, I think this depends. If there are only a few short comments below the new post, then I agree with Dysprosia.  On the other hand, if a discussion gets too long, and if there are a large number of comments below it, I favor moving it to the bottom of the page.


 * (Just mentioning that this isn't standard practice, as far as I understand what you mean. Dysprosia 23:25, 7 March 2006 (UTC))


 * Now, to address Bo's point. The polynomials are usually defined in terms of integer or real number coefficients.  If you want to use polynomials to define the integers, then you need to first define polynomials with natural number coefficients.  This is non-standard, and does not belong in this article. Rick Norwood 14:11, 7 March 2006 (UTC)


 * Rick and Oleg refer to usually without any precise reference, while I am requested to provide references. Well, the nice recursive definition of a polynomial is used in Quantum Mechanics by Sin-Itiro Tomonaga vol II. The definition is easily modified to define polynomials over a ring R by changing the condition: "1 is a polynomial" to: "the elements of R are polynomials". A polynomial over N is also a polynomial over Z, so it is not nonstandard. There are things which you don't know and which is not original research either, and there is no excuse for your deleting such things from wikipedia. Other people might benefit even if you don't. Bo Jacoby 14:42, 7 March 2006 (UTC)
 * A book on Quantum Mechanics is hardly the place to go for basic mathematical definitions. More to the point, the natural numbers are not a ring! Rick Norwood 14:57, 7 March 2006 (UTC)

Bo Jacoby, OK, so you want to talk serious mathematics. You cannot define polymomials as you say:


 * Polynomials are defined recursively: "If A,B are polynomials, the so is A+B and AB. 1 is a polynomial, and X is a polynomial". 

This is an intuitive construction, but not a rigurous one. The rigurous construction of polynomials is even more convoluted than the construction of integers from the naturals. That's why it makes no sense to use polynomials to define integers. Oleg Alexandrov (talk) 17:00, 7 March 2006 (UTC)


 * Also, according to what precedes, the first example given on Polynomials would not be one. I think among mathematicians, there is no doubt about the definition of a polynomial: it's a map of Nn (n being the number of "variables") into the ring(?) of coefficients --- or something isomorphic to this. &mdash; MFH:Talk 23:43, 7 March 2006 (UTC)

day 2 of discussion
Anyone object
 * 1) that any a in N is a polynomial ?
 * 2) that X+b (where b in N) is a polynomial ?
 * 3) that X+b=a is polynomial equation?
 * 4) that the condition (X+b=a and X+c=d have the same solution in N) defines an equivalence relation R1 in the set M={X+b=a | a,b in N} of equations of the form X+b=a?
 * 5) that a+c=b+d also defines an equivalence relation R2 in M ?
 * 6) that R1 is a restriction of R2 ?
 * 7) that the quotient set Z=M/R2 define the integers ?
 * 8) that the statement "If A,B are polynomials, then so is A+B and AB. 1 is a polynomial, and X is a polynomial" is true ?

Rick, I did not claim that the book on Quantum Mechanics is the place to go for basic mathematical definitions. I merely claim that the reference proves that this is not original research on my part. It is true that N[X] is not a ring, but it is not to the point, for we don't need a ring. The extensions of numbers N&rarr;Z&rarr;Q match the extensions of polynomials N[X]&rarr;Z[X]&rarr;Q[X].

MFH, The first example given on Polynomial is in Z[X] but not in N[X]. The elements of N[X] are polynomials just as the elements of N are numbers.


 * No, the example is in Z[x,y,z]. And N[X] is not a ring of polynomials, I think usually one requires the coefficients of polynomials to be in a ring (algebra), since you want addition and multiplication and there is no common more primitive algebraic structure than the ring featuring that. As to your numbered list above, I object to points 5,6,7: R2 is not an equiv.relation on M (I don't see any element of M occuring in R2). R1 is not a restriction of R2 (if both are equivalence relations, one could be a subset of the other, but not a restriction since thay have the same "domain"). Finally, there is no use in taking M, i.e. polynomials: the "X" is superfluous, you only use (a,b) and in fact you get Z as N²/R2 in the classical way. &mdash; MFH:Talk 03:31, 10 March 2006 (UTC)

Oleg, 'my' definition compares favorably to all the explanations in the article. If you or MFH know a rigorous (not rigurous) definition which is also comprehensable to the readers, how come it does not appear in the article ? Bo Jacoby 12:00, 8 March 2006 (UTC)


 * As far as my understanding of polynomial ring theory serves me, the first example on this article is in Z[x, y, z], not Z[x]. Dysprosia 12:07, 8 March 2006 (UTC)

You are right and I am embarassed. Bo Jacoby 12:47, 8 March 2006 (UTC)


 * agree (didn't read this before) &mdash; MFH:Talk 03:31, 10 March 2006 (UTC)


 * Bo - In Step 4 above you define an equivalence relation on M using the condition (X+b=a and X+c=d have the same solution in N). What if b is greater than a ? Then X+b=a has no solutions in N. So all {X+b=a | b>a} have the same set of solutions in N - the empty set. Does this mean that this is a single equivalence class ? I don't think this is what you intended. Gandalf61 13:41, 8 March 2006 (UTC)

I wrote what I mean and I mean what I wrote: "the same solution". Equations having no solution do not have the same solution, even if they do have the same (empty) set of solutions. However, the relation as defined is symmetric and transitive, but not reflexive, so I must specify that R1 is the reflexive closure. Thank you. Bo Jacoby 14:45, 8 March 2006 (UTC)

Bo, all this argument is because we did not go to the bottom of things. Let me ask you:


 * What is a polynomial?

Also, what is an indeterminate, X? You are avoiding this question, and you are trying to construct the integers using polynomials, which were not defined. You must construct polynomials first, please understand that. You can't just say let X be an indeterminate. Oleg Alexandrov (talk) 15:59, 8 March 2006 (UTC)


 * You are also avoiding defining the concept of solution. What is a solution? From what I know, the solutions of an equation f(x)=0 in a set X is the set of all elements in X satisfying f(x)=0. Thus defined, the equations X+2=0 and X+3=0 have the same solution in N, which is the empty set. That is the definition. I fail to see how two different equations having the same solution, the empty set, can have different solutions.


 * Unless you make it rigurous and perfectly mathematically clear what a polynomial is, and what a solution to a polynomial equation is, this discussion does not make sense. Oleg Alexandrov (talk) 16:13, 8 March 2006 (UTC)


 * Bo - let's take a concrete example - the polynomial equations X+2=1 (A), X+3=1 (B) and X+3=2 (C). None of them have solutions in N. I think you want (A) and (C) to be in the same equivalence class, and (B) to be in a different equivalence class. So - what is it about the empty set of solutions to (A) that makes it the same as the empty set of solutions to (C), but different from the empty set of solutions to (B) ? Gandalf61 16:22, 8 March 2006 (UTC)


 * In addition to agreeing strongly with Oleg and Gandalf61, let me also point out to Bo that this article is the main article on the subject of polynomials. It is not about ways to use polynomials to construct negative numbers.  That is a non-standard way to construct negative numbers, and just because somebody did it that way does not make it important enough for inclusion here. Rick Norwood 01:01, 9 March 2006 (UTC)

Oleg - The set, N[X], of polynomials is defined recursively, just as the set, N, of natural numbers is defined recursively by Peanos axioms. You are not told what a natural number is, only that the axioms apply. Basicly the axioms says that if a is in N then so is a+1, and that 1 is in N. You are not told what 1 is, and you are not told what +1 means. From the axioms, addition and multiplication are defined, and the arithmetic rules, ''a+b=b+a. (a+b)+c=a+(b+c), a(b+c)=ab+ac, ab=ba, (ab)c=a(bc), a+b=a+c => b=c, and ab=ac => b=c'' are proved. N is closed under addition and multiplication: If a,b are in N then so is a+b and ab. N is characterized by these two conditions. (1): N is closed under addition and multiplication, and (2): 1 is in N. Now define N[X] by similar conditions: (1): N[X] is closed under addition and multiplication, (2): 1 is in N[X], and (3): X is in N[X]. You are not told what X is, only that the rules apply. That is a perfectly sound definition. A solution to an equation, f(X)=g(X), is a number, a, which substituted for X in the equation produces a true statement: f(a)=g(a). A solution is not a set of numbers, but a number. The equation X+2=0 has no solution in N. Nor has the equations X+3=0 a solution in N. So they do not have the same solution. The two equations do not have different solutions. They have no solutions.

Gandalf - No, I do not want X+2=1 (A) and X+3=2 (C) to be in the same equivalence class under the relation R1, as the two equations do not have the same solution in N. But they are equivalent under the extended relation R2 because 2+2=1+3. So the two equations define the same integer, which is the solution to either in Z.

Rick - It is not nonsens what I wrote about long division. I explained the difference between exact and approximate solutions to polynomial equations, the point being that the root of a first degree polynomial like 3x-1 cannot be expressed exactly any more or any less than the roots of a fifth degree polynomial like x5-x-1. The expression x=1/3 says nothing more than 3x-1=0. I you want to know the root you must approximate: x=0.3333. This is done by long division. Your claim: "When the polynomial is of degree greater than four, it is not always possible to find exact expressions for the zeroes" is not true, because the polynomial itself is an exact expression for the roots. The descartes method is basicly useless and has but historical interest and does not belong in an introduction. Please hesitate reverting other peoples edit. Be open minded and don't reject new stuff until you understand it. The article is very bad by now. There is no proper definition, the history section is very narrow, there are no application, and a lot of the text is untrue. It need to be improved, but it cannot be improved when you just revert any change. I don't expect you to agree, but you should ask questions first and shoot afterwards. Bo Jacoby 01:54, 9 March 2006 (UTC)

part 3
You wrote:


 * The equation X+2=0 has no solution in N. Nor has the equations X+3=0 a solution in N. So they do not have the same solution. The two equations do not have different solutions. They have no solutions. 

Look, you are saying it is false that the equations have the same solution, and it is also false that they don't have the same solution. That can't be, logically. Oleg Alexandrov (talk) 03:03, 9 March 2006 (UTC)

They have the same solution set (the empty set), which I think is really what matters here. Dysprosia 04:07, 9 March 2006 (UTC)


 * Dyprosia: how would it make you feel if I said that today I agree with all you say? &mdash; MFH:Talk 03:31, 10 March 2006 (UTC)

It is logical to say that "you and I are neither dating the same girl nor different girls" if at least one of us is not dating any girl. Now substitute "dating a girl" with "having a solution". A solution is not the same thing as the set of solutions - if the set of solutions is empty, then there is no solution. The relationship (X+a=b)R1(X+c=d) does not mean that the two equations have the same set of solutions, but the relationship is true EITHER if each of the two equations has a solution and these solutions are the same, OR if a=c and b=d. (Any equation must be equivalent to itself). This relationship implies that a+d=b+c, that is: (X+a=b)R2(X+c=d). This implication does not go backwards: The equivalence relation R2 is a proper extension to R1. It contains more relationships. Let Z=M/R2 be the new set of equivalence classes of equations. There is a one-to-one correspondance between N and the subset of Z of equivalence classes of equations having the same solutions in N. A number is identified with the equivalence class of equation having that number as a solution. This identification extends the concept of number from N to Z. Bo Jacoby 10:01, 9 March 2006 (UTC)


 * But this means that you can't define integers using R1 alone - you have to use R2, so that X+2=1 and X+3=2 are equivalent under R2, even though they aren't equivalent under R1 (following your understanding of R1). In which case it is simpler to go straight to R2, and avoid R1 and polynomials and polynomial equations altogether. Your approach could, perhaps, be patched up to give a rigorous definition of the integers via polynomials with natural number coefficients, but it is becoming increasing baroque and unnatural. Oleg and others have already told you this, and yet you still seem to believe that your approach is simple and natural. I am not going to spend any more time on this discussion. Gandalf61 10:38, 9 March 2006 (UTC)

Thanks for your contributions so far. How do you explain b&minus;a with a,b in N ? To me it is a solution to the equation X+a=b. (Postulating R2 without R1 would omit this explanation). The same approach explains fractions and radicals and algebraic numbers. What explanation is natural to you ? It is your turn to be constructive. I'll then resist the temptation of being too critical. :) Bo Jacoby 11:19, 9 March 2006 (UTC)


 * At some point, Bo, you just have to accept the fact that your view that this subject belongs in this article has no support from anyone else. I appreciate your efforts to contribute to Wikipedia, but there comes a time to accept what has been said and move on.  We can agree to disagree, but one person cannot unilaterally change an article.  Rick Norwood 14:38, 9 March 2006 (UTC)

You are definining a set M={X+b=a | a,b in N}. You can't do that. You cannot define a set of equalities, an equality is not a set element. You should say the set of all pairs (X+b, a), now that's correct. Oleg Alexandrov (talk) 16:02, 9 March 2006 (UTC)


 * Oleg: you will feel less comfortable than Dysprosia, since I disagree somehow. Once you say to me what you think an equation is, I could very well consider sets of equations. Expecially polynomial equations, just like linear equations (which would be in bijection with (coordinate space)^(n+1) - you even can consider such sets of equations as vector spaces, or even as modules...) &mdash; MFH:Talk 03:31, 10 March 2006 (UTC)


 * I don't understand your point about me being not comfortable. :) Yes, you have to say what an equation is. I mean, we are trying to construct integers here, so we need to first construct all the tools needed in the process. Oleg Alexandrov (talk) 03:41, 10 March 2006 (UTC)

Rick - Every child in the whole world has to learn what an integer is, so it must be well explained in wikipedia. By now it is not well explained. My approach obviously provokes not only a lot of constructiv mathematical discussion but also some impatient dogmatic opposition. So I challenge my honored opponents to write an explanation which is understandable to the child and acceptable to Oleg and even to myself. Do you think that is easy ?

Oleg - You agree to consider sets of numbers, sets of pairs of numbers, and even sets of pairs of polynomials, but you disagree to consider sets of equations. Why is that ? Isn't an equation a mathematical object too ? The math teacher writes on the blackboard: $$\ X+a=b$$, points at it, and says: "This is an equation". An equation, like a polynomial, has several interpretations: unevaluated it is a formal mathematical object - fully evaluated it is a truth value. The equation X+a=b is unevaluated - it is not a claim that the two sides are equal polynomials. The polynomial "X+a" is also unevaluated - It is not a number, but either a function, x&rarr;(x+a), or an abstract mathematical object in its own right. The equation X+a=b may be considered a boolean function of an integer: x&rarr;(x+a=b). Here it even means (a,b)&rarr;(x&rarr;(x+a=b)) : to every pair, (a,b), there correspond a function that to a number x, tests whether x+a=b is true or not. If we stop talking about these equations and just talk about pairs then nobody will understand the meaning of the equivalence a+d=b+c. Your objection indicates that we should write fully: M = { (a,b)&rarr;(x&rarr;(x+a=b)) | a,b,x in N } rather than shorthand: M = { X+a=b | a,b in N }. Personally I prefer using leftarrow rather that rightarrow for specifying functions, f=(y&larr;x) meaning y=f(x), because y is more important that x, but that is nonstandard. In that notation the definition would be like this: M = { ((x+a=b)&larr;x)&larr;(a,b) | a,b,x in N }  Bo Jacoby 10:22, 10 March 2006 (UTC)

If you are trying to explain something to others, like above, you should use common notation, not your own notation. :) But I think I got it. Thanks.

Look, you are saying that the equations X+3=1 and X+4=2 define the same solution in N. That is wrong. Assume that the solution is the same, call it n&ge;0. By plugging it in the first equation you see that it does not satisfy the equation, contradiction. Oleg Alexandrov (talk) 16:24, 10 March 2006 (UTC)


 * Oleg, Dyprosia gave a better explanation: You can't call the solution n before showing that there is one unique solution --- in fact the solution set is here the empty set { }. (This also shows that several equations which are not "equivalent according to R_2" have the same solutions : the equation X+3=2 has the same solution (in N) than X+3=1 !). &mdash; MFH:Talk 18:37, 13 March 2006 (UTC)


 * I haven't been following this all too carefully, but if two equations have empty solution set, you can't really say they have the same solutions (since they have none), but as I mentioned, they have the same solution set. Dysprosia 21:42, 13 March 2006 (UTC)

Oleg, I agree that the equations X+3=1 and X+4=2 do not have the same solution in N. Neither of the two equations have a solution in N. So the two equations are not equivalent under the relation R1 (which demands that two equations have the same solution in N). But they are equivalent under the extended equivalence relation R2, because 3+2=1+4. So they define the same element in Z. Remember that the Z-element b&minus;a is the set of equations of the form X+c=d where a+d=b+c. Dysprosia's last remark is perfectly correct. Bo Jacoby 23:15, 13 March 2006 (UTC)


 * As Gandalf was saying right above, you should then go directly to R2, no need to complicate life with R1. You want some motivation for the construction, but your approach is the wrong way to do it. If I were to teach that thing to my class, I would spend 10 minutes ranting about the intuition behind this all, then then say "OK guys", now let us to the rigurous construction. You are trying to get to both the intuitive motivation and the rigurous construction in one shot, and if anything, you should have been convinced by now that it is utterly confusing. Oleg Alexandrov (talk) 01:03, 14 March 2006 (UTC)


 * Oleg's observation is a valid one. We have exhausted nearly half a talk page arguing about a construction that isn't really standard, lucid, verifiable, or source-able. My reading of the above is that there is a consensus for the construction not to be in the article. Let's put this one to bed, shall we? Sorry Bo. Dysprosia 06:14, 14 March 2006 (UTC)

Oleg, going directly to R2, the approach cannot be reused for constructing fractions and algebraic numbers. I was not trying to get to both the intuitive motivation and the rigorous construction in one shot. I wrote the intuitive non-rigorous section
 * http://en.wikipedia.org/w/index.php?title=Polynomial&oldid=41456817#Using_polynomials_for_extending_the_concept_of_number.

You then requested a rigorous presentation, claiming that it couldn't be made. I showed that it can be made, albeit confusing, as rigor often is.

Dysprosia, If you have a construction that is really standard, lucid, verifiable, and source-able, then write it down and let us use it. Let us not solve a solved problem. Bo Jacoby 13:42, 14 March 2006 (UTC)


 * You must choose the right tools for the job. It does not make any sense to use polynomials for the construction of the naturals, they can be bypassed and the obtained construction without polynomials is so much easier. It does not make any sense to put the problem of fractions in polynomial terms; it makes much more sense to view it as a ring theory problem (with inverse of an element).


 * The only place where it makes sense to use polynomials is the construction of algebraic numbers. Then the polynomials are natural and given.


 * We can argue about this till we drop. So let me make clear what I had said a while ago. Please do not add to Wikipedia your own deductions. Please stick to established material from books. There is a good reason for why certain constructions are preferred, and Wikipedia is the least appropriate place for discussing original ideas and interpretations. Oleg Alexandrov (talk) 16:15, 14 March 2006 (UTC)


 * We don't need one. The polynomial article will do fine without a construction of the integers (or the naturals). Dysprosia 22:18, 14 March 2006 (UTC)

I enjoyed the discussion. When your arguments were refuted, you did not change your mind regarding the conclusion, but you just invented new objections. So your argumentation is reverse, constructing arguments for your conclusion rather than deriving a conclusion from the arguments. Surely, that kind of argumentation can go on till we drop. My purpose is not to make you change your mind, because that probably can not be done anyway, but to improve the formulation. You did pinpoint weak spots in the text. By now the polynomial article is without motivation and application. That is bad. Also there is nowhere an explanation for the construction of integers, fractions and algebraic numbers in general. That is bad too. I trust that you guys will mend these defects, as you do not allow me to do it. Bo Jacoby 09:02, 15 March 2006 (UTC)


 * You make a lot of bold assertions in your above statement, many not entirely correct. In order not to spur more unnecessary discussion, let me just remark that it is not a personal matter at hand. We do not accept original research from contributors. Your construction appears to be an original one, from yourself -- even though you say you observe something in a quantum mechanics text, you did mention that you adapted this yourself. You are of course invited to contribute in any other fashion that you can, that falls within Wikipedia's policies and guidelines. Dysprosia 09:27, 15 March 2006 (UTC)

Graph of polynomials in cartesian coordinates to understand number of real roots
I've added a small paragraph which gives a graphical explanation to the concept of number of real roots allowed for a polynomial. I want to expand on it a little, but am unsure whether is would serve a useful purpose in the article, or if it would be a waste of time, and a waste of space. Also, I don't think I've explained it clearly... I understand concepts perfectly in my head, but I can't construct them into meaningful descriptions... and i'm not experienced enough with wikipedia to use a diagram to aid the concept (no to mention that it's not really a significant enough concept to warrant a picture in the article. Some feedback would be appreciated. tomohawk 09:51, 18 May 2006 (UTC)


 * I appreciate your effort, but I think this is covered already. Rick Norwood 13:21, 18 May 2006 (UTC)


 * No worries :) tomohawk 13:51, 18 May 2006 (UTC)

Motivation
Polynomials are important for a number of reasons: 1. A sum of polynomials is a polynomial 2. A product of polynomials is a polynomial 3. The derivative of a polynomial is a polynomial 4. The integral of a polynomial is a polynomial 5. Polynomials serve to approximate many functions, such as  sine, cosine, and exponential. This text from www.jsoftware.com contains motivation in a nutshell. There is presently no section on motivation in the article. Bo Jacoby 11:04, 22 June 2006 (UTC)


 * Good! Something on the reason polynomial functions are important should certainly be in the introduction.  I would add to the above, which is rather technical and only applies within mathematics, that polynomials are the first and simplest examples of functions encountered by most students, and that many natural laws are expressed in the form of polynomial functions.  The most famous example is E = mc^2.

Only addition and multiplication
I'm not sure about the latest change in the lead from
 * a polynomial is an expression in which a finite number of constants and variables are combined using only addition, subtraction, multiplication, and non-negative integer exponents (raising to a power).

to
 * a polynomial is an expression in which a finite number of constants and variables are combined using only addition and multiplication. Subtraction can be obtained by adding -1 times the subtrahend. Powers with non-negative integral exponents can be obtained by repeated multiplication.

Both are equivalent, but I think the former is simpler for the layman. We could mention the fact later that it can be generated by just addition and multiplication, just not in the lead. --Salix alba (talk) 19:04, 28 September 2006 (UTC)
 * I agree. For now I reverted to the version a month ago which is what you mention first above. Oleg Alexandrov (talk) 02:43, 29 September 2006 (UTC)
 * When I saw the edit by 70.19.63.31, it occurred to me that the definition seems arbitrary. With people throwing in subtraction and some powers, but not other powers. So I thought it would be better to limit the definition to the basics and see the other things as derived properties. After all, if we include subtraction, then why not include division in the definition? JRSpriggs 03:04, 29 September 2006 (UTC)
 * If real numbers can be added, subtracted, and multiplied by, then why can they not be used as exponents? I guess the real issue is what is the motivation for including some functions and not others? JRSpriggs 03:12, 29 September 2006 (UTC)

We don't get to decide what we would like the word "polynomial" to mean. The word has a well established meaning, and that meaning does not allow non-natural real number powers or division by a variable. Rick Norwood 13:24, 9 December 2006 (UTC)

Is (1+x)(1-x) a polynomial?
The anons recent edit to the lead raises an interesting distinction between a purely formal definition of a polynomial as the sum of monomials, and equations which can be reduced to that form. For example Stewart Galois Theory defines polynomials as a formal sum. Hence the formal expression $$(1+x)(1-x)$$ is not a polynomial but its expanded form $$1-x^2$$ is. Should we make this clear in the lead? --Salix alba (talk) 01:03, 5 November 2006 (UTC)


 * Formal polynomials are a special case. In the usual use of the word, and expression equivalent to a polynomial is a polynomial, just as 6 and 2*3 are both natural numbers. Rick Norwood 13:26, 9 December 2006 (UTC)
 * It is a good point. Of cause both 1&minus;x2 and (1+x)(1&minus;x) are polynomials, and if the definition indicates otherwise, then the definition must be changed. I suggest the following recursive definition of a polynomial over a ring R.
 * The constant polynomials: Any member of R is in R[x].
 * The variable: x is in R[x].
 * Closed against addition and multiplication: If A and B are in R[x], then so are the sum A+B and the product AB.
 * This definition does not refer to monomials and so is simpler and easier to understand. Bo Jacoby 17:37, 11 December 2006 (UTC)

Roots
The statement: "Analytic solutions of the roots of a polynomial in terms of its coefficients are possible using only the standard arithmetic operations and the extraction of roots only if the degree of the polynomial is four or lower", is not correct. A root of the polynomial x5&minus;a may be expressed using standard arithmetic operations and the extraction of roots, even if the degree is greater than four. (x is a fifth root of a). Please transform the statement into a correct one, or delete it. Bo Jacoby 00:09, 8 December 2006 (UTC)
 * I agree it was not well phrased. I tried to fix that. Oleg Alexandrov (talk) 01:21, 8 December 2006 (UTC)
 * Well done Oleg. Thanks. Bo Jacoby 17:17, 11 December 2006 (UTC)