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Diagonal polynomials for small dimensions

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Abstract

HereR andN denote respectively the real numbers and the nonnegative integers. Also 0 <n εN, ands(x) =x 1+...+x n when x = (x 1,...,x n) εR n. Adiagonal function of dimensionn is a mapf onN n (or any larger set) that takesN n bijectively ontoN and, for all x, y inN n, hasf(x) <f(y) whenevers(x) <s(y). We show that diagonalpolynomials f of dimensionn all have total degreen and have the same terms of that degree, so that the lower-degree terms characterize any suchf. We call two polynomialsequivalent if relabeling variables makes them identical. Then, up to equivalence, dimension two admits just one diagonal polynomial, and dimension three admits just two.

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References

  1. G. Cantor, Ein Beitrag zur Mannigfaltigkeitslehre,J. Keine Angew. Math. (Crelle's Journal),84 (1878), 242–258.

    Article  Google Scholar 

  2. R. Fueter and G. Pólya, Rationale Abzählung der Gitterpunkte,Vierteljschr. Naturforsch. Gesellsch. Zurich,58 (1923), 380–386.

    Google Scholar 

  3. J. S. Lew and A. L. Rosenberg, Polynomial indexing of integer lattice points, I and II,J. Number Theory,10 (1978), 192–214 and 215–243.

    Article  MathSciNet  MATH  Google Scholar 

  4. L. B. Morales and J. S. Lew, An enlarged family of packing polynomials on multidimensional lattices,Math. Systems Theory, this issue, pp. 293–303.

  5. G. Pólya and G. Szegö,Aufgaben und Lehrsatze aus der Analysis, Springer-Verlag, New York, 1964; translated asProblems and Theorems in Analysis, Vol. 2, Springer-Verlag, New York, 1976, Problem 243.

    Google Scholar 

  6. T. Skolem, Über die Zurückführbarkeit einiger durch Rekursionen definierter Relationen auf “Arithmetische” (1937), in T. Skolem (ed.),Selected Works in Logic, Universitetsforlaget, Oslo, 1970.

    Google Scholar 

  7. T. Skolem, The development of recursive arithmetic (1947), in T. Skolem (ed.),Selected Works in Logic, Universitetsforlaget, Oslo, 1970.

    Google Scholar 

  8. C. Smoryński,Logical Number Theory: An Introduction, Vol. 1, Springer-Verlag, Berlin, 1991, pp. 14–43.

    Google Scholar 

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Authors and Affiliations

  1. Mathematical Sciences Department, IBM Research Division, T. J. Watson Research Center, 10598, Yorktown Heights, NY, USA

    J. S. Lew

  2. IIMAS, Universidad Nacional Autónoma de México, C.P. 01000, Apdo. Postal 20-726, México D.F., Mexico

    L. B. Morales & A. Sánchez-Flores

Authors
  1. J. S. Lew
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  2. L. B. Morales
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  3. A. Sánchez-Flores
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Lew, J.S., Morales, L.B. & Sánchez-Flores, A. Diagonal polynomials for small dimensions. Math. Systems Theory 29, 305–310 (1996). https://doi.org/10.1007/BF01201282

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  • DOI: https://doi.org/10.1007/BF01201282

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