Talk:Pisot–Vijayaraghavan number

An anonymous contributor put this article in Category:algebraic integers which does not exist. I wonder, should one create such a category, or just put this article in Category:Algebraic number theory. Thanks. Oleg Alexandrov 03:53, 7 Mar 2005 (UTC)

Algebraic number theory is not the same as the theory of algebraic numbers; it is really the theory of algebraic number fields. I suggest a Category:Algebraic numbers, since, for example, there are results of diophantine approximation that talk about individual numbers. Charles Matthews 11:51, 7 Mar 2005 (UTC)

What's the command for "infinity" in TeX ? I spelled out "infinity" in letters in the formula lim_n x^n - a_n in the article, someone might want to fix that. --FvdP 22:49, 23 December 2005 (UTC)

\infty -- 216.137.30.161 (talk) 06:24, 9 May 2013 (UTC)

Pisot numbers

The tendency among people in the field seems to be towards calling these simply Pisot numbers; hence I mention this name for them in the first sentence. Gene Ward Smith 21:29, 11 April 2006 (UTC)

Does "x^n tends to integers" imply "x is Pisot" ?

I'm a bit puzzled over this statement "The converse holds: if x is a real number > 1 and there is a sequence of integers an so that $\lim _{n\to \infty }x^{n}-a_{n}=0$, then x is a Pisot-Vijayaraghavan number." It seems to be an open question not so long ago, so it may be still unknown. If it has been recently resolved, it would be nice to include a reference.

(written by User:Fedja who forgot to sign)

You may be right. AFAIR I am the one who added that "the converse holds". I was sure I read this result somewhere, about 10 years ago or more, as proven. Not so sure now. Since I'm no specialist I may have been wrong all along. The close but more general "Pisot-Vijayaraghavan conjecture" looks still open indeed. --FvdP 20:04, 20 July 2006 (UTC)

OK, as I could find no confirmation of it (though no clear refutation either), I removed the dubious statement. --FvdP 21:20, 17 August 2006 (UTC)

The table of small PV numbers

Let S denote the set of PV numbers. I have thought of some questions about the structure of S. Anybody know what has been published about them?

1) Let B denote the set of PV numbers that are less than the golden ratio. I find that the first 9 numbers of B have discriminants -23, -283, 1609, -31, 37253, 3857, -691055, 29077,and 4477. I wonder whether every number in B has a symmetric group as its Galois group. The first 9 numbers do, usually because the discriminant has an unsquared prime factor.

2) What kind of numbers are limit numbers in S? Have they been completely classified?

Scott Tillinghast, Houston TX 02:18, 7 March 2007 (UTC) Revised Scott Tillinghast, Houston TX 07:33, 10 March 2007 (UTC) Revised Scott Tillinghast, Houston TX 08:52, 8 April 2007 (UTC) Revised Scott Tillinghast, Houston TX (talk) 05:36, 29 January 2009 (UTC)

Not encyclopedic? Why would the discriminant of an algebraic number be any less important than a 20-place decimal representation? Discriminants may indicate that 2 polynomials generate non-isomorphic algebraic number fields and often identify Galois groups. In algebraic number theory a discriminant is a pretty standard piece of information about an algebraic number or the polynomial that defines it.

As for sources, most of these discriminants are found in http://www.math.uni-duesseldorf.de/~klueners/minimum/minimum.html, although some were my own computations.

The table has one exceptional polynomial of the 6th degree. In the interest of space I would leave blank its unconsolidated version. Also, doing so emphasizes that it is exceptional. Scott Tillinghast, Houston TX (talk) 14:52, 27 September 2010 (UTC)

The hidden sentence at the end of this section about generators of real fields is covered in the section 'Elementary properties' and can be deleted. Scott Tillinghast, Houston TX (talk) 15:03, 27 September 2010 (UTC)

Multiple problems

Since this article gets good attention, I will wait to see if someone else wants to change the following:

1) A certain section reads

The smallest Pisot-Vijayaraghavan number is the unique real solution of $x^{3}-x-1=0$, (approximatively 1.324718). [some lines]

The decic :$(x-1)(x^{10}+x^{9}-x^{7}-x^{6}-x^{5}-x^{4}-x^{3}+x+1)=0$ has root x = 1.176280... which is the aforementioned smallest known Salem number

So the smallest known Salem number is 1.1762808.. but it is NOT the plastic number (I will take that claim off silver number in a moment).

2) I was glad to see this section. I never knew how that degree ten polynomial (better name for general use then decic IMHO) came from. Keep the section! but

The general approach is to take the minimal polynomial P(x) of a Pisot-Vijayaraghavan number and its reciprocal, P(1/x), to form the equation

is very confusing. If P=P(x) is a polynomial of degree n then the reciprocal polynomial is P*=P*(x)=x^nP(1/x). If P is the minimal polynomial of a number α, then the minimal polynomial of 1/α is P*. NOW one could explain this OR (since it is a general method but only used here in one case) just say something like

$x^{n}(x^{3}-x-1)=-(x^{3}+x^{2}-1)$ where (by the way) x^3+x^2-1 is the minimal polynomial of the reciprocal of the plastic number

3) Aha, as I recall, the plastic number is the smallest Pisot number ---which does not come from a self reciprocal polynomial --- (that being P=P*)

4) Are the things in the list all units? (I don't know) Is it mentioned because a previous incarnation of the article made some false claim about unit? If so, with the claim gone no need to say so. --Gentlemath (talk) 18:09, 14 March 2009 (UTC)

OK, I need to read more carefully. I would correct my point 1) above but maybe that is bad Wikipedia practice? Maybe Salem number should enter the introduction. I think maybe it is true that 1.176... is the smallest Salem or Pisot number with an non-reciprocal minimal polynomial.

5) Perhaps a unit with respect to Pisot numbers is one whose minimal polynomial ends in +1 or -1. Could that be? If so, it should be said.--Gentlemath (talk) 20:03, 14 March 2009 (UTC)

Yes, a unit is an algebraic integer whose minimal polynomial ends in +1 or -1. Scott Tillinghast, Houston TX (talk) 15:14, 27 September 2010 (UTC)

Expert tag: The 38

The article lists "the 38" PV numbers less than φ implying there are exactly 38 of them. Computations I have done indicate that the root of xⁿ(x²-x-1) + 1 which is >1 is always a PV number when n>1 and in the limit these numbers apporach φ from below. Is there a citation for the number 38 that I can check? Normally I just do a fact tag but the statement in the article seems wrong.--RDBury (talk) 10:17, 14 September 2009 (UTC)

There are indeed infinitely many PV numbers < φ (and φ is the smallest limit point and isolated within the set of limit points), see Jacques Dufresnoy, Charles Pisot: Étude de certaines fonctions méromorphes bornées sur le cercle unité. Application à un ensemble fermé d'entiers algébriques, Annales scientifiques de l'École Normale Supérieure 72, 1955, pp. 69–92 --91.32.121.61 (talk) 12:18, 24 September 2009 (UTC)

However: the article lists "the 38" PV numbers less than the decimal fraction 1.618. I’d considerably shorten the list, though, e.g. keeping the ten smallest PV numbers. --91.32.121.61 (talk) 12:31, 24 September 2009 (UTC)

Thanks for clearing that up; somehow I kept reading 1.618 as φ (conditioning I suppose). The facts you mention should certainly be added to the article and I'll try to do that when I get a chance.--RDBury (talk) 12:28, 25 September 2009 (UTC)

I agree, it seems rather arbitrary to look below 1.618. So I was WP:bold and cut it down to 10 as suggested. I also cut down the other list of quadratic roots to 10. Cheers, — sligocki (talk) 19:46, 11 November 2009 (UTC)

Also, I added a number of {{citation needed}} tags, the claims didn't seem self-evident (to me), can someone source them? I'll try to give the sources a look myself. Cheers, — sligocki (talk)

I believe Bertin et al give a proof that all the PV numbers that are less than the golden section are roots of one exceptional sextic polynomial or of the 2 infinite families of polynomials. When I make a trip to the library I can make a footnote. Scott Tillinghast, Houston TX (talk) 17:32, 27 September 2010 (UTC)

The citation in Bertin et. al. shows how to find all the PV numbers less than 1.6. It does not list decimal values. Scott Tillinghast, Houston TX (talk) 02:58, 24 October 2010 (UTC)

Uploaded Axel Thue's 1912 paper to archive.org

Hello, this article references a 1912 paper by Axel Thue. I was interested in reading it, and managed to get a copy and have uploaded it to archive.org and wish to add it to the references but am not sure how. I included a link to the resource on archive.org: https://archive.org/details/uber-eine-eigenschaft-thue Bloouup (talk) 01:27, 10 February 2021 (UTC)