Editing Salem number

{{Use American English|date = March 2019}}
{{Short description|Type of algebraic integer}}

[[File:Salem numbers.svg|thumb|300px|Plot of the roots of Lehmer's [[polynomial]], with the corresponding Salem number near <math>x=1.17628</math> in gold.]]

In [[mathematics]], a '''Salem number''' is a [[real number|real]] [[algebraic integer]] <math>\alpha > 1</math> whose [[conjugate roots]] all have [[absolute value]] no greater than 1, and at least one of which has absolute value exactly&nbsp;1. Salem numbers are of interest in [[Diophantine approximation]] and [[harmonic analysis]]. They are named after [[Raphaël Salem]].

== Properties ==

Because it has a root of absolute value 1, the [[minimal polynomial (field theory)|minimal polynomial]] for a Salem number must be a [[reciprocal polynomial]]. This implies that <math>1/\alpha</math> is also a root, and that all other roots have absolute value exactly one. As a consequence α must be a [[unit (ring theory)|unit]] in the [[ring (mathematics)|ring]] of algebraic integers, being of [[field norm|norm]] 1.

Every Salem number is a [[Perron number]] (a real algebraic number greater than one all of whose conjugates have smaller absolute value).

== Relation with Pisot–Vijayaraghavan numbers==

The smallest known Salem number is the largest real root of '''Lehmer's polynomial'''<!-- redirects here --> (named after [[Derrick Henry Lehmer]])

:<math>P(x) = x^{10} + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1,</math>

which is about <math>x = 1.17628</math>: it is [[conjecture]]d that it is indeed the smallest Salem number, and the smallest possible [[Mahler measure]] of an [[irreducible polynomial|irreducible]] non-[[cyclotomic polynomial|cyclotomic]] polynomial.<ref>Borwein (2002) p.16</ref>

Lehmer's polynomial is a factor of the shorter [[degree of a polynomial|degree]]-12 polynomial,

:<math>Q(x) = x^{12} - x^7 - x^6 - x^5 + 1,</math>

all twelve roots of which satisfy the relation<ref>D. Bailey and D. Broadhurst, [http://crd-legacy.lbl.gov/~dhbailey/dhbpapers/ladder.pdf A Seventeenth Order Polylogarithm Ladder]</ref>

:<math>x^{630}-1 = \frac{(x^{315}-1)(x^{210}-1)(x^{126}-1)^2(x^{90}-1)(x^{3}-1)^3(x^{2}-1)^5(x-1)^3 }{(x^{35}-1)(x^{15}-1)^2(x^{14}-1)^2(x^{5}-1)^6\,x^{68}}</math>

Salem numbers can be constructed from [[Pisot–Vijayaraghavan number]]s. To recall, the smallest of the latter is the unique real root of the [[cubic polynomial]],

:<math>x^3 - x - 1,</math>

known as the ''[[plastic ratio]]'' and approximately equal to 1.324718. This can be used to generate a family of  Salem numbers including the smallest one found so far.  The general approach is to take the minimal polynomial <math>P(x)</math> of a Pisot–Vijayaraghavan number and its [[reciprocal polynomial]], <math>P^*(x)</math>, and solve the equation,

:<math>x^n P(x) = \pm P^{*}(x)</math>

for [[integer]] <math>n</math> above a bound. Subtracting one side from the other, factoring, and disregarding trivial factors will then yield the minimal polynomial of certain Salem numbers. For example, using the negative case of the above,

:<math>x^n(x^3 - x - 1) = -(x^3 + x^2 - 1)</math>

then for <math>n=8</math>, this factors as,

:<math>(x-1)(x^{10} + x^9 -x^7 -x^6 -x^5 -x^4 -x^3 +x +1) = 0</math>

where the [[Decic equation|decic]] is Lehmer's polynomial. Using higher <math>n</math> will yield a family with a root approaching the plastic ratio. This can be better understood by taking <math>n</math>th roots of both sides,

:<math>x(x^3 - x - 1)^{1/n} = \pm (x^3 + x^2 - 1)^{1/n} </math>

so as <math>n</math> goes higher, <math>x</math> will approach the solution of <math>x^3 - x - 1=0</math>.  If the positive case is used, then <math>x</math> approaches the plastic ratio from the opposite direction.  Using the minimal polynomial of the next smallest Pisot–Vijayaraghavan number gives

:<math>x^n(x^4 - x^3 - 1) = -(x^4 + x - 1),</math>

which for <math>n=7</math> factors as

:<math>(x-1)(x^{10} - x^6 - x^5 - x^4 + 1) = 0,</math>

a decic not generated in the previous and has the root <math>x=1.216391\ldots</math> which is the 5th smallest known Salem number. As <math>n\to\infty</math>, this family in turn tends towards the larger real root of <math>x^4 - x^3 - 1 = 0</math>.

== References ==
{{reflist}}
* {{cite book | last=Borwein | first=Peter | authorlink=Peter Borwein | title=Computational Excursions in Analysis and Number Theory | series=CMS Books in Mathematics | publisher=[[Springer-Verlag]] | year=2002 | isbn=0-387-95444-9 | zbl=1020.12001 }} Chap. 3.
* {{springer|first=David|last= Boyd|title=Salem number|id=s/s120010}}
* {{cite web|author=M.J. Mossinghoff|url=http://www.cecm.sfu.ca/~mjm/Lehmer/lists/SalemList.html|title=Small Salem numbers|accessdate=2016-01-07}}
* {{cite book | last=Salem | first=R. | authorlink=Raphaël Salem | title=Algebraic numbers and Fourier analysis | series=Heath mathematical monographs | location=Boston, MA | publisher=[[D. C. Heath and Company]] | year=1963 | zbl=0126.07802 }}

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[[Category:Algebraic numbers]]