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Fueter–Pólya theorem

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

Introduction

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

is a bijective 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.[2]

Statement

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If is a real quadratic polynomial in two variables whose restriction to is a bijection from to then it is

or

Proof

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

Fueter–Pólya conjecture

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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

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

The sum of these binomial coefficients yields a polynomial of degree in variables. This is just one of at least inequivalent packing polynomials for dimensions.[6]

References

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  1. ^ G. Cantor: Ein Beitrag zur Mannigfaltigkeitslehre, J. Reine Angew. Math., Band 84 (1878), Pages 242–258
  2. ^ Rudolf Fueter, Georg Pólya: Rationale Abzählung der Gitterpunkte, Vierteljschr. Naturforsch. Ges. Zürich 68 (1923), Pages 380–386
  3. ^ Craig Smoryński: Logical Number Theory I, Springer-Verlag 1991, ISBN 3-540-52236-0, Chapters I.4 and I.5: The Fueter–Pólya Theorem I/II
  4. ^ M. A. Vsemirnov, Two elementary proofs of the Fueter–Pólya theorem on pairing polynomials. St. Petersburg Math. J. 13 (2002), no. 5, pp. 705–715. Correction: ibid. 14 (2003), no. 5, p. 887.
  5. ^ P. Chowla: On some Polynomials which represent every natural number exactly once, Norske Vid. Selsk. Forh. Trondheim (1961), volume 34, pages 8–9
  6. ^ Sánchez Flores, Adolfo (1995). "A family of diagonal polynomial orders of ". Order. 12 (2): 173–187. doi:10.1007/BF01108626.