Talk:Pisot–Vijayaraghavan number

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

message: 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)
thread_title: Pisot numbers

message: 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)
replies: text: 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)
thread_title: Does "x^n tends to integers" imply "x is Pisot" ?

message: 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)
thread_title: The table of small PV numbers

message: 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)
thread_title:  Multiple problems  

message: 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)
replies: text: 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)

text: 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)
replies: text: 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)
thread_title:  Expert tag: The 38 

message: 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)
thread_title:  Uploaded Axel Thue's 1912 paper to archive.org