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.
Similar content being viewed by others
References
G. Cantor, Ein Beitrag zur Mannigfaltigkeitslehre,J. Keine Angew. Math. (Crelle's Journal),84 (1878), 242–258.
R. Fueter and G. Pólya, Rationale Abzählung der Gitterpunkte,Vierteljschr. Naturforsch. Gesellsch. Zurich,58 (1923), 380–386.
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.
L. B. Morales and J. S. Lew, An enlarged family of packing polynomials on multidimensional lattices,Math. Systems Theory, this issue, pp. 293–303.
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.
T. Skolem, Über die Zurückführbarkeit einiger durch Rekursionen definierter Relationen auf “Arithmetische” (1937), in T. Skolem (ed.),Selected Works in Logic, Universitetsforlaget, Oslo, 1970.
T. Skolem, The development of recursive arithmetic (1947), in T. Skolem (ed.),Selected Works in Logic, Universitetsforlaget, Oslo, 1970.
C. Smoryński,Logical Number Theory: An Introduction, Vol. 1, Springer-Verlag, Berlin, 1991, pp. 14–43.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01201282