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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The number of certain integral polynomials and nonrecursive sets of integers, Part 1

Authors: Tamás Erdélyi and Harvey Friedman
Journal: Trans. Amer. Math. Soc. 357 (2005), 999-1011
MSC (2000): Primary 41A17; Secondary 30B10, 26D15
Published electronically: October 5, 2004
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Abstract | References | Similar Articles | Additional Information

Abstract: Given $r > 2$, we establish a good upper bound for the number of multivariate polynomials (with as many variables and with as large degree as we wish) with integer coefficients mapping the ``cube'' with real coordinates from $[-r,r]$ into $[-t,t]$. This directly translates to a nice statement in logic (more specifically recursion theory) with a corresponding phase transition case of 2 being open. We think this situation will be of real interest to logicians. Other related questions are also considered. In most of these problems our main idea is to write the multivariate polynomials as a linear combination of products of scaled Chebyshev polynomials of one variable.

References [Enhancements On Off] (What's this?)

  • [BE] P. Borwein and T. Erdélyi, Polynomials and Polynomial Inequalities, Springer-Verlag, New York, 1995. MR 97e:41001
  • [LGM] Lorentz, G.G., M. von Golitschek, and Y. Makovoz, Constructive Approximation, Advanced Problems, Springer, Berlin, 1996. MR 97k:41002

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Additional Information

Tamás Erdélyi
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843

Harvey Friedman
Affiliation: Department of Mathematics, The Ohio State University, 231 West Eighteenth Avenue, Columbus, Ohio 43210

Keywords: Multivariate polynomials, integer coefficients, Chebyshev polynomials, orthogonality, Parseval formula
Received by editor(s): July 15, 2003
Published electronically: October 5, 2004
Article copyright: © Copyright 2004 American Mathematical Society