In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a Pisot–Vijayaraghavan number, also called simply a Pisot number or a PV number, is a
real algebraic integer
In algebraic number theory, an algebraic integer is a complex number that is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial (a polynomial whose leading coefficient is 1) whose coefficients ...
greater than 1, all of whose
Galois conjugates are less than 1 in
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
. These numbers were discovered by
Axel Thue
Axel Thue (; 19 February 1863 – 7 March 1922) was a Norwegian mathematician, known for his original work in diophantine approximation and combinatorics.
Work
Thue published his first important paper in 1909.
He stated in 1914 the so-called w ...
in 1912 and rediscovered by
G. H. Hardy
Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis. In biology, he is known for the Hardy–Weinberg principle, a basic principle of pop ...
in 1919 within the context of
Diophantine approximation
In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria.
The first problem was to know how well a real number can be approximated ...
. They became widely known after the publication of
Charles Pisot's dissertation in 1938. They also occur in the uniqueness problem for
Fourier series
A Fourier series () is an Series expansion, expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems ...
.
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
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
:
measuring the distance from its consecutive powers to the nearest
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
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
Cubic may refer to:
Science and mathematics
* Cube (algebra), "cubic" measurement
* Cube, a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex
** Cubic crystal system, a crystal system w ...
irrationality known as the
plastic ratio. Much is known about the
accumulation points of ''S''. The smallest of them is the
golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if
\fr ...
.
Definition and properties
An algebraic integer of degree ''n'' is a root ''α'' of an
irreducible monic polynomial
In algebra, a monic polynomial is a non-zero univariate polynomial (that is, a polynomial in a single variable) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1. That is to say, a monic polynomial is one ...
''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
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
numbers of absolute value less than 1, so that they lie strictly inside the
unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
in the
complex plane
In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...
, then ''α'' is called a Pisot number, Pisot–Vijayaraghavan number, or simply PV number. For example, the
golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if
\fr ...
, ''φ'' ≈ 1.618, is a real
quadratic integer
In number theory, quadratic integers are a generalization of the usual integers to quadratic fields. A complex number is called a quadratic integer if it is a root of some monic polynomial (a polynomial whose leading coefficient is 1) of degree tw ...
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
Elementary properties
* Every integer greater than 1 is a PV number. Conversely, every
rational
Rationality is the quality of being guided by or based on reason. In this regard, a person acts rationally if they have a good reason for what they do, or a belief is rational if it is based on strong evidence. This quality can apply to an ...
PV number is an integer greater than 1.
* If α is an
irrational
Irrationality is cognition, thinking, talking, or acting without rationality.
Irrationality often has a negative connotation, as thinking and actions that are less useful or more illogical than other more rational alternatives. The concept of ...
PV number whose minimal polynomial ends in ''k'' then α is greater than , ''k'', .
* If α is a PV number then so are its powers α
''k'', for all positive integer exponents ''k''.
* Every real
algebraic number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a ...
K of degree ''n'' contains a PV number of degree ''n''. This number is a field generator. The set of all PV numbers of degree ''n'' in K is closed under multiplication.
* Given an upper bound ''M'' and degree ''n'', there are only
finitely many of PV numbers of degree ''n'' that are less than ''M''.
* Every PV number is a
Perron number (a real algebraic number greater than one all of whose conjugates have smaller absolute value).
Diophantine properties
The main interest in PV numbers is due to the fact that their powers have a very "biased" distribution (mod 1). If ''α'' is a PV number and ''λ'' is any algebraic integer in the
field then the sequence
:
where , , ''x'', , denotes the distance from the real number ''x'' to the nearest integer, approaches 0 at an exponential rate. In particular, it is a square-summable sequence
and its terms converge to 0.
Two converse statements are known: they characterize PV numbers among all real numbers and among the algebraic numbers (but under a weaker Diophantine assumption).
* Suppose ''α'' is a real number greater than 1 and ''λ'' is a non-zero real number such that
::
:Then ''α'' is a Pisot number and ''λ'' is an algebraic number in the field
(Pisot's theorem).
* Suppose ''α'' is an algebraic number greater than 1 and ''λ'' is a non-zero real number such that
::
:Then ''α'' is a Pisot number and ''λ'' is an algebraic number in the field
.
A longstanding Pisot–Vijayaraghavan problem asks whether the assumption that ''α'' is algebraic can be dropped from the last statement. If the answer is affirmative, Pisot's numbers would be characterized ''among all real numbers'' by the simple convergence of , , ''λα''
''n'', , to 0 for some auxiliary real ''λ''. It is known that there are only
countably many numbers ''α'' with this property. The problem is to decide whether any of them is
transcendental.
Topological properties
The set of all Pisot numbers is denoted ''S''. Since Pisot numbers are algebraic, the set ''S'' is countable. Raphael Salem proved that this set is
closed: it contains all its
limit point
In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x contains a point of S other than x itself. A ...
s. His proof uses a constructive version of the main diophantine property of Pisot numbers:
[Salem (1963) p.13] given a Pisot number ''α'', a real number ''λ'' can be chosen so that 0 < ''λ'' ≤ ''α'' and
:
Thus the ''ℓ''
2 norm of the sequence , , ''λα''
''n'', , can be bounded by a uniform constant independent of ''α''. In the last step of the proof, Pisot's characterization is invoked to conclude that the limit of a sequence of Pisot numbers is itself a Pisot number.
Closedness of ''S'' implies that it has a
minimal element
In mathematics, especially in order theory, a maximal element of a subset S of some preordered set is an element of S that is not smaller than any other element in S. A minimal element of a subset S of some preordered set is defined dually as an ...
.
Carl Siegel proved that it is the positive root of the equation (
plastic constant) and is isolated in ''S''. He constructed two sequences of Pisot numbers converging to the golden ratio ''φ'' from below and asked whether ''φ'' is the smallest limit point of ''S''. This was later proved by Dufresnoy and Pisot, who also determined all elements of ''S'' that are less than ''φ''; not all of them belong to Siegel's two sequences. Vijayaraghavan proved that ''S'' has infinitely many limit points; in fact, the sequence of
derived sets
:
does not terminate. On the other hand, the intersection
of these sets is
empty, meaning that the
Cantor–Bendixson rank of ''S'' is ''ω''. Even more accurately, the
order type
In mathematics, especially in set theory, two ordered sets and are said to have the same order type if they are order isomorphic, that is, if there exists a bijection (each element pairs with exactly one in the other set) f\colon X \to Y su ...
of ''S'' has been determined.
The set of
Salem numbers, denoted by ''T'', is intimately related with ''S''. It has been proved that ''S'' is contained in the set ''T of the limit points of ''T''.
[Salem (1963) p.30] It has been
conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...
d that the
union of ''S'' and ''T'' is closed.
[Salem (1963) p. 31]
Quadratic irrationals
If
is a
quadratic irrational
In mathematics, a quadratic irrational number (also known as a quadratic irrational or quadratic surd) is an irrational number that is the solution to some quadratic equation with rational coefficients which is irreducible over the rational numb ...
there is only one other conjugate,
, obtained by changing the sign of the
square root
In mathematics, a square root of a number is a number such that y^2 = x; in other words, a number whose ''square'' (the result of multiplying the number by itself, or y \cdot y) is . For example, 4 and −4 are square roots of 16 because 4 ...
in
from
:
or from
:
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
or
, and are satisfied in the second case exactly when
and either
or
.
Thus, the first few quadratic irrationals that are PV numbers are:
Powers of PV-numbers
Pisot–Vijayaraghavan numbers can be used to generate
almost integers: the ''n''th power of a Pisot number approaches integers as ''n'' grows. For example,
:
Since
and
differ by only
:
is extremely close to
:
Indeed
:
Higher powers give correspondingly better rational approximations.
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; this follows from an application of
Newton's identities. When ''x'' is a Pisot number, the ''n''th powers of the other conjugates tend to 0 as ''n'' tends to infinity. Since the sum is an integer, the distance from ''x
n'' to the nearest integer tends to 0 at an exponential rate.
Small Pisot numbers
All Pisot numbers that do not exceed the
golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if
\fr ...
''φ'' have been determined by Dufresnoy and Pisot. The table below lists ten smallest Pisot numbers in increasing order.
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,
[Bertin et al., p. 133.] with the exception of
:
are factors of either
:
or
:
The first polynomial is divisible by ''x''
2 − 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 ''x''
''n'' gives expressions that approach ''x''
2 − ''x'' − 1 as ''n'' grows very large and have zeros that
converge to ''φ''. A complementary pair of polynomials,
:
and
:
yields Pisot numbers that approach φ from above.
Two-dimensional
turbulence
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to laminar flow, which occurs when a fluid flows in parallel layers with no disruption between ...
modeling using
logarithmic spiral
A logarithmic spiral, equiangular spiral, or growth spiral is a self-similarity, self-similar spiral curve that often appears in nature. The first to describe a logarithmic spiral was Albrecht Dürer (1525) who called it an "eternal line" ("ewi ...
chains with
self-similarity defined by a constant scaling factor can be reproduced with some small Pisot numbers.
References
*
* Chap. 3.
*
*
*
*
*
*
*
External links
''Pisot number'' Encyclopedia of Mathematics
*
{{DEFAULTSORT:Pisot-Vijayaraghavan Number
Algebraic numbers