Pisot-Vijayaraghavan number

In mathematics, a Pisot-Vijayaraghavan number, also called simply a Pisot number or a PV number, is an algebraic integer α which is real and exceeds 1, but such that its conjugate elements are all less than 1 in absolute value.

For example, if $alpha,$ is a quadratic irrational there is only one other conjugate: $alpha\text{'},$, obtained by changing the sign of the square root in $alpha,$; from

:$alpha\; =\; a\; +\; b\; sqrt\; d$

with "a" and "b" both integers, or in other cases both half an odd integer, we get

:$alpha\text{'}\; =\; a\; -\; b\; sqrt\; d$

The conditions are then:$alpha\; >\; 1,$

and :

This condition is satisfied by the golden ratio φ. We have

:$varphi\; =\; frac\{1\; +\; sqrt\; 5\}\{2\}\; >\; 1$

and

:$varphi\text{'}\; =\; frac\{1\; -\; sqrt\; 5\}\; 2\; =\; frac\{-1\}varphi\; .$

The general condition was investigated by G. H. Hardy in relation with a problem of diophantine approximation. This work was followed up by Tirukkannapuram Vijayaraghavan (1902–1955), an Indian mathematician from the Madras region who came to Oxford to work with Hardy in the mid-1920s. The same condition also occurs in some problems on Fourier series, and was later investigated by Charles Pisot. The name now commonly used comes from both of those authors.

Pisot-Vijayaraghavan numbers can be used to generate almost integers: the "n"th power of a Pisot number approaches integers as "n" approaches infinity. For example, consider powers of $phi,$, such as $phi^\{21\}\; =\; 24476.0000409,$. The effect can be even more pronounced for Pisot-Vijayaraghavan numbers generated from equations of higher degree.

This property stems from the fact that for each "n", the sum of "n"th powers of an algebraic integer "x" and its conjugates is exactly an integer; when "x" is a Pisot number, the n-th powers of the (other) conjugates tend to 0 as n tends to infinity.

The lowest Pisot-Vijayaraghavan number is the unique real solution of $x^3\; -\; x\; -\; 1,$, known as the "plastic number" (approximatively 1.324718).

The lowest accumulation point of the set of Pisot-Vijayaraghavan numbers is the golden ratio $varphi\; =\; frac\{1\; +\; sqrt\; 5\}\{2\}\; approx\; 1.618033$. The set of all Pisot-Vijayaraghavan numbers is nowhere dense because it is a closed and countable set.

Table of Pisot numbers

Here are the 38 Pisot numbers less than 1.618, in increasing order.

The number $2+sqrt\; 2$ is a PV number that is not a unit, since it satisfies the equation x²-4x+2=0.

Every real algebraic number field contains a PV number that generates this field. In quadratic and cubic fields it is not hard to find a unit that is a PV number.

ee also

* Salem number

External links

* [http://eom.springer.de/p/p120130.htm "Pisot number"] , Encyclopedia of Mathematics
*

References

*cite book | author=M.J. Bertin | coauthors=A. Decomps-Guilloux, M. Grandet-Hugot, M. Pathiaux-Delefosse, J.P. Schreiber | title=Pisot and Salem Numbers | publisher=Birkhäuser | year=1992 | isbn=3764326484
*cite book | author=Peter Borwein | 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 Chap. 3.
*cite journal | author=D.W. Boyd | title=Pisot and Salem numbers in intervals of the real line | journal=Math. Comp. | volume=32 | year=1978 | pages=1244–1260 | doi=10.2307/2006349
*