Fueter–Pólya theorem

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The Fueter–Pólya theorem, first proved by Rudolf Fueter and George Pólya, states that the only quadratic pairing functions are the Cantor polynomials.

Introduction[]

In 1873, Georg Cantor showed that the so-called Cantor polynomial^[1]

${\displaystyle P(x,y):={\frac {1}{2}}((x+y)^{2}+3x+y)={\frac {x^{2}+2xy+3x+y^{2}+y}{2}}=x+{\frac {(x+y)(x+y+1)}{2}}={\binom {x}{1}}+{\binom {x+y+1}{2}}}$

is a bijective mapping from ${\displaystyle \mathbb {N} ^{2}}$ to ${\displaystyle \mathbb {N} }$. 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 ${\displaystyle P(0,0)=0}$. He then wrote to Pólya, who showed the theorem does not require this condition.^[2]

Statement[]

If ${\displaystyle P}$ is a real quadratic polynomial in two variables whose restriction to ${\displaystyle \mathbb {N} ^{2}}$ is a bijection from ${\displaystyle \mathbb {N} ^{2}}$ to ${\displaystyle \mathbb {N} }$ then it is

${\displaystyle P(x,y):={\frac {1}{2}}((x+y)^{2}+3x+y)}$

or

${\displaystyle P(x,y):={\frac {1}{2}}((y+x)^{2}+3y+x).}$

Proof[]

The original proof is surprisingly difficult, using the Lindemann–Weierstrass theorem to prove the transcendence of ${\displaystyle e^{a}}$ for a nonzero algebraic number ${\displaystyle a}$.^[3] In 2002, M. A. Vsemirnov published an elementary proof of this result.^[4]

Fueter–Pólya conjecture[]

The theorem states that the Cantor polynomial is the only quadratic pairing polynomial of ${\displaystyle \mathbb {N} ^{2}}$ and ${\displaystyle \mathbb {N} }$. The conjecture is that these are the only such pairing polynomials.

Higher dimensions[]

A generalization of the Cantor polynomial in higher dimensions is as follows:^[5]

${\displaystyle P_{n}(x_{1},\ldots ,x_{n})=\sum _{k=1}^{n}{\binom {k-1+\sum _{j=1}^{k}x_{j}}{k}}=x_{1}+{\binom {x_{1}+x_{2}+1}{2}}+\cdots +{\binom {x_{1}+\cdots +x_{n}+n-1}{n}}}$

The sum of these binomial coefficients yields a polynomial of degree ${\displaystyle n}$ in ${\displaystyle n}$ variables. This is just one of at least ${\displaystyle (n-1)!}$ inequivalent packing polynomials for ${\displaystyle n}$ dimensions.^[6]

References[]