Editing Pisot–Vijayaraghavan number (section)

==Small Pisot numbers==

All Pisot numbers that do not exceed the [[golden ratio]] ''φ'' have been determined by Dufresnoy and Pisot. The table below lists ten smallest Pisot numbers in increasing order.<ref>{{citation
 | last1 = Dufresnoy | first1 = J.
 | last2 = Pisot | first2 = Ch.
 | journal = Annales Scientifiques de l'École Normale Supérieure
 | language = French
 | mr = 0072902
 | pages = 69–92
 | title = Etude de certaines fonctions méromorphes bornées sur le cercle unité. Application à un ensemble fermé d'entiers algébriques
 | url = http://www.numdam.org/item?id=ASENS_1955_3_72_1_69_0
 | volume = 72
 | year = 1955| doi = 10.24033/asens.1030
 }}. The smallest of these numbers are listed in numerical order on p. 92.</ref>

<div class="overflowbugx" style="overflow-x:auto;">
{| class="wikitable"
|-
!   !! Value !! Root of... !! Root of...
|-
| 1 || {{val|1.32471795724474602596|end=...}} {{OEIS2C|A060006}} ([[plastic ratio]]) || <math>x(x^2-x-1)+(x^2-1)</math> || <math>x^3-x-1</math>
|-
| 2 || {{val|1.38027756909761411567|end=...}} {{OEIS2C|A086106}} || <math>x^2(x^2-x-1)+(x^2-1)</math> ||<math>x^4-x^3-1</math>
|-
| 3 || {{val|1.44326879127037310762|end=...}} {{OEIS2C|A228777}} || <math>x^3(x^2-x-1)+(x^2-1)</math> ||<math>x^5-x^4-x^3+x^2-1</math>
|-
| 4 || {{val|1.46557123187676802665|end=...}} {{OEIS2C|A092526}} ([[supergolden ratio]]) || <math>x^3(x^2-x-1)+1</math> ||<math>x^3-x^2-1</math>
|-
| 5 || {{val|1.50159480353908736637|end=...}} {{OEIS2C|A293508}} || <math>x^4(x^2-x-1)+(x^2-1)</math> ||<math>x^6-x^5-x^4+x^2-1</math>
|-
| 6 || {{val|1.53415774491426691543|end=...}} {{OEIS2C|A293509}} || <math>x^4(x^2-x-1)+1</math> ||<math>x^5-x^3-x^2-x-1</math>
|-
| 7 || {{val|1.54521564973275524325|end=...}} {{OEIS2C|A293557}} || <math>x^5(x^2-x-1)+(x^2-1)</math> ||<math>x^7-x^6-x^5+x^2-1</math>
|-
| 8 || {{val|1.56175206772029729470|end=...}} {{OEIS2C|A374002}} || <math>x^3(x^3-2x^2+x-1)+(x-1)(x^2+1)</math> ||<math>x^6-2x^5+x^4-x^2+x-1</math>
|-
| 9 || {{val|1.57014731219605436291|end=...}} {{OEIS2C|A293506}} || <math>x^5(x^2-x-1)+1</math> ||<math>x^5-x^4-x^2-1</math>
|-
| 10 || {{val|1.57367896839351698877|end=...}} {{OEIS2C|A374003}} || <math>x^6(x^2-x-1)+(x^2-1)</math> ||<math>x^8-x^7-x^6+x^2-1</math>
|}
</div>

Since these PV numbers are less than 2, they are all units: their minimal polynomials end in 1 or −1.
<!-- this needs to be put in proper place or removed
Every real algebraic number field contains a PV number that generates this field. In real quadratic and cubic fields it is not hard to find a unit that is a PV number.
-->
The polynomials in this table,<ref name=":0">Bertin et al., p. 133.</ref> with the exception of

: <math>x^6 - 2x^5 + x^4 - x^2 + x - 1,</math>

are factors of either

:<math>x^n(x^2 - x - 1) + 1</math>
or
:<math>x^n(x^2 - x - 1) + (x^2 - 1).</math>

The first polynomial is divisible by ''x''<sup>2</sup>&nbsp;−&nbsp;1 when ''n'' is odd and by ''x''&nbsp;−&nbsp;1 when ''n'' is [[parity (mathematics)|even]]. It has one other real zero, which is a PV number. Dividing either polynomial by ''x''<sup>''n''</sup> gives expressions that approach ''x''<sup>2</sup>&nbsp;−&nbsp;''x''&nbsp;−&nbsp;1 as ''n'' grows very large and have zeros that [[limit of a sequence|converge]] to ''φ''.  A complementary pair of polynomials,

:<math>x^n(x^2 - x - 1) - 1</math>
and
:<math> x^n(x^2-x-1) - (x^2-1)\,</math>

yields Pisot numbers that approach φ from above.

Two-dimensional [[turbulence]] modeling using [[logarithmic spiral]] chains with [[self-similarity]] defined by a constant scaling factor can be reproduced with some small Pisot numbers.<ref>{{cite journal |url=https://journals.aps.org/pre/abstract/10.1103/PhysRevE.100.043113 |author1=Ö. D. Gürcan |author2= Shaokang Xu|author3=P. Morel |title=Spiral chain models of two-dimensional turbulence |journal=Physical Review E |volume=100 |date=2019 |issue=4 |page=043113 |doi=10.1103/PhysRevE.100.043113 |pmid=31770954 |arxiv=1903.09494 }}</ref>