The Number of Certain Integral Polynomials
نویسندگان
چکیده
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. In some private communications, Harvey Friedman raised the following problem: given r > 2, give an 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 variables from [−r, r] into [−t, t]. Robin Pemantle has established a rough upper bound. Here, utilizing Chebyshev polynomials, we establish a reasonably good upper bound. Namely, in this paper we prove our main result and some related ones, applications of which in recursion theory are given by Harvey Friedman in a separate article. We think that the two papers are so closely related that we decided to publish them in the same journal. The Main Result Theorem 1. Let r > 2. 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 variables from [−r, r] into [−t, t] is at most (2t + 1) 2 ( t log 2 t)/((log 2)(log(r/2)) )t2 ≤ exp(c t log t) , where the constant c depends only on r. In the above theorem, and throughout the paper, log without a specified base means the natural logarithm with the base e. To prove the theorem we need a few lemmas. 1991 Mathematics Subject Classification. Primary: 41A17, Secondary: 30B10, 26D15.
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تاریخ انتشار 2004