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)
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