The Fueter–Pólya theorem, first proved by
Rudolf Fueter and
George Pólya
George Pólya (; ; December 13, 1887 – September 7, 1985) was a Hungarian-American mathematician. He was a professor of mathematics from 1914 to 1940 at ETH Zürich and from 1940 to 1953 at Stanford University. He made fundamental contributi ...
, states that the only
quadratic polynomial
pairing functions are the Cantor
polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
s.
Introduction
In 1873,
Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor ( ; ; – 6 January 1918) was a mathematician who played a pivotal role in the creation of set theory, which has become a foundations of mathematics, fundamental theory in mathematics. Cantor establi ...
showed that the so-called Cantor polynomial
:
is a
bijective
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
mapping from
to
.
The polynomial given by swapping the variables is also a pairing function.
Fueter was investigating whether there are other quadratic polynomials with this property, and concluded that this is not the case assuming
. He then wrote to Pólya, who showed the theorem does not require this condition.
Statement
If
is a real quadratic polynomial in two variables whose
restriction to
is a bijection from
to
then it is
:
or
:
Proof
The original proof is surprisingly difficult, using the
Lindemann–Weierstrass theorem to prove the transcendence of
for a nonzero algebraic number
.
In 2002, M. A. Vsemirnov published an elementary proof of this result.
Fueter–Pólya conjecture
The theorem states that the Cantor polynomial is the only ''quadratic'' pairing polynomial of
and
. The conjecture is that these are the only such pairing polynomials, of any degree.
Higher dimensions
A generalization of the Cantor polynomial in higher dimensions is as follows:
:
The sum of these
binomial coefficients
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the te ...
yields a polynomial of degree
in
variables. This is just one of at least
inequivalent packing polynomials for
dimensions.
References
{{DEFAULTSORT:Fueter-Polya theorem
Mathematical theorems
Number theory