Talk:Transcendence theory

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Not worth an edit war; but it is inexact to call x in P(x) a kind of formal variable. Charles Matthews 22:39, 16 Nov 2004 (UTC)

My problem is that P(x)=0 on its own, without x quantified, doesn't really mean anything. Your point is that it stands for the map x -> P(x), and in some contexts we might use this confusion. For example we say the function sin x. However the tone of this article is pretty formal and I think it is necessary to say P(x)=0 for all x, or perhaps better to use the notation later in the article P=0. Or could say P(x) is identically zero (rather more old fashioned I think). Billlion 07:32, 17 Nov 2004 (UTC)

Well, no, strictly, P is not a mapping but a formal expression. And the assertion is that it is the constant 0 (as formal expression, also); which is the notation for the polynomial with all its coefficients zero. Charles Matthews 08:25, 17 Nov 2004 (UTC)

Now we are in to some interesting pedantry! In the line above P(e)=0 clearly refers to the evaluation of P at e, so we are identifying the formal expression with a function evaluated at a real number. I am now more convinced P=0 is best, as P is clearly an object in the module of integer coeff polynomials. Billlion 09:51, 17 Nov 2004 (UTC)

Not to rude, but I think that the description in the first sentence on the Transcendental number article is best. The one I edited in here is very similar. Furthermore, this article needs some cleanup, in my opinion. Look below:

The quantitative approach asks one to find lower bounds
P(e) > F(A,d)
depending on a bound A of the coefficients of P and its degree, that apply to all P ≠ 0.

I, as a reader of the article, have some questions about this:

  • What is the function F?
  • The lower bounds of what exactly?
  • What is the function F?
  • I think A is a number that is greater than the magnitude (absolute value) of any coeffient of the polynomial function P, right?
  • What is the function F?
  • d is the polynomial degree of P, right?
  • What is the function F?
Do you see the point I am trying to make clear?
  • Also, where are the references? I can't check your work or believe anything unless there is some reading that I am inclined to read. (preferably there be an online one so that people do not have to go to the library, but at least 1 book so that it is more verifiable)
  • EulerGamma 21:29, 11 September 2006 (UTC)

[edit] Overlap with Transcendental number article

I'd love to help get this article into reasonable shape, but it seems inevitable to me that it is going to overlap a lot with the transcendental number article. I'm definitely not suggesting the two be merged but it'd be useful to get some suggestions on what to put into which article Chenxlee (talk) 20:02, 13 February 2008 (UTC)