The number of certain integral polynomials and nonrecursive sets of integers, Part 1
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- by Tamás Erdélyi and Harvey Friedman PDF
- Trans. Amer. Math. Soc. 357 (2005), 999-1011 Request permission
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
- Peter Borwein and Tamás Erdélyi, Polynomials and polynomial inequalities, Graduate Texts in Mathematics, vol. 161, Springer-Verlag, New York, 1995. MR 1367960, DOI 10.1007/978-1-4612-0793-1
- George G. Lorentz, Manfred v. Golitschek, and Yuly Makovoz, Constructive approximation, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 304, Springer-Verlag, Berlin, 1996. Advanced problems. MR 1393437, DOI 10.1007/978-3-642-60932-9
Additional Information
- Tamás Erdélyi
- Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
- Email: terdelyi@math.tamu.edu
- Harvey Friedman
- Affiliation: Department of Mathematics, The Ohio State University, 231 West Eighteenth Avenue, Columbus, Ohio 43210
- MR Author ID: 69465
- Email: friedman@math.ohio-state.edu
- Received by editor(s): July 15, 2003
- Published electronically: October 5, 2004
- © Copyright 2004 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 357 (2005), 999-1011
- MSC (2000): Primary 41A17; Secondary 30B10, 26D15
- DOI: https://doi.org/10.1090/S0002-9947-04-03631-1
- MathSciNet review: 2110429