In mathematics, a Pisot–Vijayaraghavan number, also called simply a Pisot number or a PV number, is a real algebraic integer greater than 1, all of whose Galois conjugates are less than 1 in absolute value. These numbers were discovered by Axel Thue in 1912 and rediscovered by G. H. Hardy in 1919 within the context of diophantine approximation. They became widely known after the publication of Charles Pisot's dissertation in 1938. They also occur in the uniqueness problem for Fourier series. Tirukkannapuram Vijayaraghavan and Raphael Salem continued their study in the 1940s. Salem numbers are a closely related set of numbers.

A characteristic property of PV numbers is that their powers approach integers at an exponential rate. Pisot proved a remarkable converse: if α > 1 is a real number such that the sequence

${\displaystyle \|\alpha ^{n}\|}$

measuring the distance from its consecutive powers to the nearest integer is square-summable, or ℓ ², then α is a Pisot number (and, in particular, algebraic). Building on this characterization of PV numbers, Salem showed that the set S of all PV numbers is closed. Its minimal element is a cubic irrationality known as the plastic ratio. Much is known about the accumulation points of S. The smallest of them is the golden ratio.

Definition and properties

An algebraic integer of degree n is a root α of an irreducible monic polynomial P(x) of degree n with integer coefficients, its minimal polynomial. The other roots of P(x) are called the conjugates of α. If α > 1 but all other roots of P(x) are real or complex numbers of absolute value less than 1, so that they lie strictly inside the circle |x| = 1 in the complex plane, then α is called a Pisot number, Pisot–Vijayaraghavan number, or simply PV number. For example, the golden ratio, φ ≈ 1.618, is a real quadratic integer that is greater than 1, while the absolute value of its conjugate, −φ⁻¹ ≈ −0.618, is less than 1. Therefore, φ is a Pisot number. Its minimal polynomial is x² − x − 1.

Quadratic irrationals

If ${\displaystyle \alpha \,}$ is a quadratic irrational there is only one other conjugate: ${\displaystyle \alpha '\,}$, obtained by changing the sign of the square root in ${\displaystyle \alpha \,}$ from

${\displaystyle \alpha =a+{\sqrt {D}}{\text{ to }}\alpha '=a-{\sqrt {D}}\,}$

or from

${\displaystyle \alpha ={\frac {a+{\sqrt {D}}}{2}}{\text{ to }}\alpha '={\frac {a-{\sqrt {D}}}{2}}.\,}$

Here a and D are integers and in the second case a is odd and D is congruent to 1 modulo 4.

The required conditions are α > 1 and −1 < α' < 1. These are satisfied in the first case exactly when a > 0 and either ${\displaystyle (a-1)^{2}<D<a^{2}}$ or ${\displaystyle a^{2}<D<(a+1)^{2}}$. These are satisfied in the second case exactly when ${\displaystyle a>0}$ and either ${\displaystyle (a-2)^{2}<D<a^{2}}$ or ${\displaystyle a^{2}<D<(a+2)^{2}}$.

Thus, the first few quadratic irrationals that are PV numbers are:

Value	Root of...	Numerical value
${\displaystyle {\frac {1+{\sqrt {5}}}{2}}}$	${\displaystyle x^{2}-x-1}$	1.618033... OEIS: A001622 (the golden ratio)
${\displaystyle 1+{\sqrt {2}}\,}$	${\displaystyle x^{2}-2x-1}$	2.414213... OEIS: A014176 (the silver ratio)
${\displaystyle {\frac {3+{\sqrt {5}}}{2}}}$	${\displaystyle x^{2}-3x+1}$	2.618033... OEIS: A104457
${\displaystyle 1+{\sqrt {3}}\,}$	${\displaystyle x^{2}-2x-2}$	2.732050... OEIS: A090388
${\displaystyle {\frac {3+{\sqrt {13}}}{2}}}$	${\displaystyle x^{2}-3x-1}$	3.302775... OEIS: A098316 (the third metallic mean)
${\displaystyle 2+{\sqrt {2}}\,}$	${\displaystyle x^{2}-4x+2}$	3.414213...
${\displaystyle {\frac {3+{\sqrt {17}}}{2}}}$	${\displaystyle x^{2}-3x-2}$	3.561552.. OEIS: A178255.
${\displaystyle 2+{\sqrt {3}}\,}$	${\displaystyle x^{2}-4x+1}$	3.732050... OEIS: A019973
${\displaystyle {\frac {3+{\sqrt {21}}}{2}}}$	${\displaystyle x^{2}-3x-3}$	3.791287...OEIS: A090458
${\displaystyle 2+{\sqrt {5}}\,}$	${\displaystyle x^{2}-4x-1}$	4.236067... OEIS: A098317 (the fourth metallic mean)

Powers of PV-numbers

Pisot–Vijayaraghavan numbers can be used to generate almost integers: the nth power of a Pisot number approaches integers as n grows. For example,

${\displaystyle (3+{\sqrt {10}})^{6}=27379+8658{\sqrt {10}}=54757.9999817\dots \approx 54758-{\frac {1}{54758}}.}$

Since ${\displaystyle 27379\,}$ and ${\displaystyle 8658{\sqrt {10}}\,}$ differ by only ${\displaystyle 0.0000182\dots ,\,}$

${\displaystyle {\frac {27379}{8658}}=3.162277662\dots }$

is extremely close to

${\displaystyle {\sqrt {10}}=3.162277660\dots .}$

Indeed

${\displaystyle \left({\frac {27379}{8658}}\right)^{2}=10+{\frac {1}{8658^{2}}}.}$

Higher powers give correspondingly better rational approximations.

This property stems from the fact that for each n, the sum of nth powers of an algebraic integer x and its conjugates is exactly an integer; this follows from an application of Newton's identities. When x is a Pisot number, the nth powers of the other conjugates tend to 0 as n tends to infinity. Since the sum is an integer, the distance from xⁿ to the nearest integer tends to 0 at an exponential rate.

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 the increasing order.^[8]

	Value	Root of...	Root of...
1	1.3247179572447460260 OEIS: A060006 (plastic ratio)	${\displaystyle x(x^{2}-x-1)+(x^{2}-1)}$	${\displaystyle x^{3}-x-1}$
2	1.3802775690976141157 OEIS: A086106	${\displaystyle x^{2}(x^{2}-x-1)+(x^{2}-1)}$	${\displaystyle x^{4}-x^{3}-1}$
3	1.4432687912703731076 OEIS: A228777	${\displaystyle x^{3}(x^{2}-x-1)+(x^{2}-1)}$	${\displaystyle x^{5}-x^{4}-x^{3}+x^{2}-1}$
4	1.4655712318767680267 OEIS: A092526 (supergolden ratio)	${\displaystyle x^{3}(x^{2}-x-1)+1}$	${\displaystyle x^{3}-x^{2}-1}$
5	1.5015948035390873664 OEIS: A293508	${\displaystyle x^{4}(x^{2}-x-1)+(x^{2}-1)}$	${\displaystyle x^{6}-x^{5}-x^{4}+x^{2}-1}$
6	1.5341577449142669154 OEIS: A293509	${\displaystyle x^{4}(x^{2}-x-1)+1}$	${\displaystyle x^{5}-x^{3}-x^{2}-x-1}$
7	1.5452156497327552432 OEIS: A293557	${\displaystyle x^{5}(x^{2}-x-1)+(x^{2}-1)}$	${\displaystyle x^{7}-x^{6}-x^{5}+x^{2}-1}$
8	1.5617520677202972947	${\displaystyle x^{3}(x^{3}-2x^{2}+x-1)+(x-1)(x^{2}+1)}$	${\displaystyle x^{6}-2x^{5}+x^{4}-x^{2}+x-1}$
9	1.5701473121960543629 OEIS: A293506	${\displaystyle x^{5}(x^{2}-x-1)+1}$	${\displaystyle x^{5}-x^{4}-x^{2}-1}$
10	1.5736789683935169887	${\displaystyle x^{6}(x^{2}-x-1)+(x^{2}-1)}$	${\displaystyle x^{8}-x^{7}-x^{6}+x^{2}-1}$

Since these PV numbers are less than 2, they are all units: their minimal polynomials end in 1 or −1. The polynomials in this table,^[9] with the exception of

${\displaystyle x^{6}-2x^{5}+x^{4}-x^{2}+x-1,}$

are factors of either

${\displaystyle x^{n}(x^{2}-x-1)+1\,}$

or

${\displaystyle x^{n}(x^{2}-x-1)+(x^{2}-1).\ }$

The first polynomial is divisible by x² − 1 when n is odd and by x − 1 when n is even. It has one other real zero, which is a PV number. Dividing either polynomial by xⁿ gives expressions that approach x² − x − 1 as n grows very large and have zeros that converge to φ. A complementary pair of polynomials,

${\displaystyle x^{n}(x^{2}-x-1)-1}$

and

${\displaystyle x^{n}(x^{2}-x-1)-(x^{2}-1)\,}$

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.^[10]

References

• M.J. Bertin; A. Decomps-Guilloux; M. Grandet-Hugot; M. Pathiaux-Delefosse; J.P. Schreiber (1992). Pisot and Salem Numbers. Birkhäuser. ISBN 3-7643-2648-4.
• Borwein, Peter (2002). Computational Excursions in Analysis and Number Theory. CMS Books in Mathematics. Springer-Verlag. ISBN 0-387-95444-9. Zbl 1020.12001. Chap. 3.
• Boyd, David W. (1978). "Pisot and Salem numbers in intervals of the real line". Math. Comp. 32: 1244–1260. doi:10.2307/2006349. ISSN 0025-5718. Zbl 0395.12004.
• Cassels, J. W. S. (1957). An introduction to Diophantine approximation. Cambridge Tracts in Mathematics and Mathematical Physics. Vol. 45. Cambridge University Press. pp. 133–144.
• Hardy, G. H. (1919). "A problem of diophantine approximation". J. Indian Math. Soc. 11: 205–243.
• Ö. D. Gürcan; Shaokang Xu; P. Morel (2019). "Spiral chain models of two-dimensional turbulence". Physical Review E. 100. arXiv:1903.09494. doi:10.1103/PhysRevE.100.043113.
• Pisot, Charles (1938). "La répartition modulo 1 et nombres algébriques". Ann. Sc. Norm. Super. Pisa II. Ser. 7 (in French): 205–248. Zbl 0019.15502.
• Salem, Raphaël (1963). Algebraic numbers and Fourier analysis. Heath mathematical monographs. Boston, MA: D. C. Heath and Company. Zbl 0126.07802.
• Thue, Axel (1912). "Über eine Eigenschaft, die keine transzendente Größe haben kann". Christiania Vidensk. selsk. Skrifter. 2 (20): 1–15. JFM 44.0480.04.

External links

• Pisot number, Encyclopedia of Mathematics
• Terr, David & Weisstein, Eric W. "Pisot Number". MathWorld.