Talk:Vieta's formulas

Vieta or Viete[edit]

The more commonly-used name by far is Vieta's formulas not Viète's formulas. (You can verify that on MathWorld and elsewhere). How to rename this page?
I gave it a severe rewrite for improving clarity
It is not massively important that this extends to complex $\lbrace x_{k}\rbrace$
Noone cares much about the abstract algebra generalization to rings (remove? or move to bottom)

Smcinerney (talk) 12:38, 3 August 2008 (UTC)[reply]

The guy is called Viète, so the formulas are Viète's formulas. Oleg Alexandrov (talk) 15:13, 3 August 2008 (UTC)[reply]

Oleg, that's inaccurate. I'm repeating the fact that the name by which the formulas are commonly known is Vieta's formulas, regardless whether the mathematician's name is more commonly given as Viète. Here are two citations to prove this:

This is analogous to there being multiple anglicizations of Chebyshev/Tschebysheff/etc. The formulas should be listed under the name by which they are actually known. Smcinerney (talk) 19:43, 3 August 2008 (UTC)[reply]

Not that I really care much one way or the other, but Viète would have written in Latin, and would have transliterated his name as Vieta. Not sure where this leaves the debate, though. siℓℓy rabbit (talk) 01:37, 6 August 2008 (UTC)[reply]

I also know them as Vieta's formulae. (In German, actually, it is the same phenomenon: de:François Viète and de:Satzgruppe_von_Vieta.) Jakob.scholbach (talk) 08:00, 6 August 2008 (UTC)[reply]

I'm with Oleg here: Viète's formulas. CRGreathouse (t | c) 04:35, 7 August 2008 (UTC)[reply]

The original poster is correct. Wikipedia's manual of style requires that we use the name most commonly recognised in English, which is "Vieta's formulas", irrespective of what the man called himself. JamesBWatson (talk) 12:30, 4 July 2011 (UTC)[reply]

This is a silly matter, but I and my colleagues (at a fairly well-ranked university in the United States) refer to these as Viète's formulas. I also am somewhat involved with the history of math and the few times I've heard them referenced I believe it has been as Viète's formulas, although I could possibly just be mis-remembering as it's entirely possible that the formulas didn't actually come up. At any rate, this might be an occasion where those who are very close to the matter refer to them in one way while they are commonly referred to by a different name. Just a thought. — Preceding unsigned comment added by 108.2.60.94 (talk) 22:48, 8 February 2014 (UTC)[reply]

I think the last sentence summarizes it well. In English at least, the tradition has been to use Vieta, and Viete is an overcorrection (outside historical or biographic contexts), like saying "Hero's formula" with Greek pronunciation, instead than Heron's formula. 73.89.25.252 (talk) 04:52, 13 June 2020 (UTC)[reply]

Multiple name for a page[edit]

How to have multiple names for a page? Or how to have something like a redirecting?

This formula in high school is sometimes referred to as 'the formula for the sum and product of the roots' or the likes. User:Appleuseryu — Preceding undated comment added 18:22, 28 April 2015 (UTC)[reply]

In Wikipedia, a page has a unique name. However WP:Redirects allows to get the same page by typing several names. For example Viète formulas, Vieta formulas, and many others redirect to this page. You are asking for a more explicit title. Unfortunately the one you suggest is too long and convenient only for quadratic polynomials. The only alternative name that I am able to suggest is elementary symmetric function of the roots. As elementary symmetric function redirects to elementary symmetric polynomial, I have edited this article for linking it to here. D.Lazard (talk) 21:49, 28 April 2015 (UTC)[reply]

Slightly confusing notation perhaps[edit]

The argument to the polynomial is given as x, per convention, but the roots are also given as x1, x2 etc. That threw me for a minute as I though there was some link between them I was missing. Would using r1, r2 etc. for the roots be more mnemonic, and surely less confusing?

Also the Funkhauser link doesn't seem to work. 92.29.233.168 (talk) 22:22, 8 October 2018 (UTC)[reply]

I have fixed the reference.

It is not a convention, but a common usage to denote

x

the indeterminate (or variable) of a polynomial. The link between

x

and the roots of the polynomial is immediately given by the definition of a root: the roots are the values of

x

for which the polynomial is zero. As they are finite in number, it is natural to name them

x_{1},\ldots

It is possible to give them other names, but this could be confusing, as, by doing this, the link between the variable and its interesting values would be hidden. D.Lazard (talk) 08:41, 9 October 2018 (UTC)[reply]

Makes perfect sense, thanks! 2.98.220.93 (talk) 09:20, 9 October 2018 (UTC)[reply]

Formulas for the monic case are the point, un-normalized case is a simple corollary.[edit]

Having the extra factor of (1/a) in all the formulas is annoying. I think most people who learned this material think of it in terms of the elementary symmetric functions being the coefficients (up to sign) of the _normalized_ polynomial, rather than having the normalization built in to each formula so it can apply to the "general" case. To handle the general situation one normalizes the polynomial. 73.89.25.252 (talk) 22:27, 11 June 2020 (UTC)[reply]

What do you mean by "to handle"? Is that "to prove" or to "use"? It is sure that to prove the formulas, it is simpler to prove first the monic case, and then to deduce the general case as a corolary. But for using the formula in a context where the polynomials are not given as monic, it is much simpler to use the general formulas directly than to convert polynomials to their monic form: this requires either a possibly confusing and error prone change of notation, or the introduction of new notations for the coefficients of the monic case. In both cases, this makes the mathematical discourse more complicate and thus less clear. D.Lazard (talk) 09:21, 12 June 2020 (UTC)[reply]

To use, teach, prove, remember, analyze, etc.

The usual procedure (and one followed for many other purposes with polynomial equations P(x)=0) is to first do the monic case and then normalize as needed. I think this article should first state the monic case, and then the "general" one as a corollary. 73.89.25.252 (talk) 03:50, 13 June 2020 (UTC)[reply]

It might be also worth reminding somewhere that although a polynomial changes when rescaled, the equation defining roots, P(x)=0, is scale invariant. It seems that some clarification of terminology could be relevant: are "roots" something attached to an equation (i.e. a scheme, variety, or ideal), or to a polynomial. Logically maybe the former but usage is ambiguous. 73.89.25.252 (talk) 03:57, 13 June 2020 (UTC)[reply]

What seems you usual is not necessarily usual for other users of Wikipedia. Even for teaching: I have never seen the quadratic formula taught by presenting first the case

a=1,

and then proving the general case by normalization. D.Lazard (talk) 08:21, 13 June 2020 (UTC)[reply]

This is precisely what is done in completing the square, solving the cubic equation (it is rare to present the formula beyond the special case of x^3+px+q), and in classic books such as Euler's algebra or Burnside's theory of equations (pp.35-36). Would there be a problem in first stating the monic case in this article? 73.89.25.252 (talk) 14:47, 13 June 2020 (UTC)[reply]

Another example. Dickson, Elementary Theory of Equations, does not give the formula for the non-normalized case. He states the monic Vieta formulas, and then that in the general case one can normalize the polynomial and adjust the formula accordingly. See chapter VI pp.55-56. 73.89.25.252 (talk) 15:22, 13 June 2020 (UTC)[reply]

@D.Lazard: Any comment on the examples, and whether to first give the formula for the normalized case? 73.89.25.252 (talk) 19:46, 12 July 2020 (UTC)[reply]

Cardano-Vieta[edit]

In Spanish, these are es:Relaciones de Cardano-Vieta, I suppose that after Girolamo Cardano. What was the role of Cardano in these formulas? Why is he recognized only in Spanish? --Error (talk) 15:50, 13 August 2020 (UTC)[reply]