Salem number

In mathematics, a real algebraic integer α > 1 is a Salem number if all its conjugate roots have absolute value no greater than 1, and at least one has absolute value exactly 1. Salem numbers are of interest in Diophantine approximation and harmonic analysis. They are named after Raphaël Salem.

1 Properties
2 Relation with Pisot–Vijayaraghavan numbers
3 See also
4 References

Properties

Because it has a root of absolute value 1, the minimal polynomial for a Salem number must be reciprocal. This implies that 1/|α| is also a root, and that all other roots have absolute value exactly one. As a consequence α must be a unit in the ring of algebraic integers, being of norm 1.

Relation with Pisot–Vijayaraghavan numbers

The smallest known Salem number is the largest real root of Lehmer's polynomial (named after Derrick Henry Lehmer)

$x^{10} %2B x^9 -x^7 -x^6 -x^5 -x^4 -x^3 %2Bx %2B1 \, ,$

which is about 1.17628.

The smallest Pisot–Vijayaraghavan number is the unique real root of the cubic polynomial

$x^3 - x - 1,$

known as the plastic number and approximately equal to 1.324718. This polynomial can be used to generate a family of Salem numbers including the smallest one found so far (x ≈ 1.176280) by Lehmer. The general approach is to take the minimal polynomial P(x) of a Pisot–Vijayaraghavan number and its reciprocal polynomial, P*(x), to form the equation,

$x^n P(x) = \pm P^{*}(x) \,$

for integral n above a bound. Subtracting one side from the other, factoring, and disregarding trivial factors yields the minimal polynomial of certain Salem numbers. For example, using the negative case of the above,

$x^n(x^3-x-1) = -(x^3%2Bx^2-1)$

then for n = 8, this factors as,

$(x-1)(x^{10} %2B x^9 -x^7 -x^6 -x^5 -x^4 -x^3 %2Bx %2B1) = 0$

where the decic has the root x ≈ 1.176280 which is the aforementioned smallest known Salem number given by Lehmer. Using higher n will yield a family with a root approaching the plastic number. This can be better understood by taking nth roots of both sides,

$x(x^3-x-1)^{1/n} = -(x^3%2Bx^2-1)^{1/n}$

so as n goes higher, x will approach the solution of x³ − x − 1 = 0 (as well as the reciprocal for even degree equations). If the positive case is used, then x approaches the plastic number from the opposite direction. Using the minimal polynomial of the next smallest Pisot–Vijayaraghavan number gives,

$x^n (x^4-x^3-1) = -(x^4%2Bx-1)$

which for n = 7 factors as,

$(x-1)(x^{10} -x^6 -x^5 -x^4 %2B1) = 0$

a decic not generated in the previous and has the root x = 1.216391... which is the 5th smallest known Salem number. As n → infinity, this family in turn tends towards the larger real root of x⁴ − x³ − 1 = 0.

References

Borwein, Peter (2002). Computational Excursions in Analysis and Number Theory. CMS Books in Mathematics. Springer-Verlag. ISBN 0-387-95444-9. Chap. 3.
Boyd, David (2001), "Salem number", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1556080104, http://www.encyclopediaofmath.org/index.php?title=s/s120010
M.J. Mossinghoff. "Small Salem numbers". http://www.math.ucla.edu/~mjm/lc/lists/SalemList.html. Retrieved 2007-01-21.

Salem number

Contents

Properties

Relation with Pisot–Vijayaraghavan numbers

See also

References