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NatNon022204 - The number of certain integral polynomials...

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The number of certain integral polynomials and nonrecursive sets of integers, part 2 Harvey M. Friedman Department of Mathematics Ohio State University [email protected] February 22, 2004 Abstract We present some examples of mathematically natural nonrecursive sets of integers and relations on integers by combining results from Part 1, recursion theory, and from the negative solution to Hilbert°s 10th Problem ([3], [1], and [2]). 1 Introduction We present some examples of mathematically natural nonrecursive sets of inte- gers and multivariate relations on integers. The usual examples of nonrecursive sets and relations on the integers involve mathematically unnatural codings of ±nite objects as integers, and may also involve models of computation and formal languages. A very important mathematically natural example of a nonrecursive set of ±nite objects comes from the work on Hilbert°s 10th Problem (see [2]). This is the set of all integral polynomials that have an integral zero. This example avoids models of computation and formal languages. However, it does not read- ily provide a mathematically natural nonrecursive set of integers. For example, one can form the nonrecursive set of integer codes of integral polynomials that have an integral zero. However, the coding of integral polynomials as integers destroys the mathematical naturalness of the example. Our approach is to use the work on Hilbert°s 10th Problem together with [3]. In particular we show that the following two sets are nonrecursive (see Corollary 26). f n 2 Z + : ( 9 P 2 IPOLY )( n = max ( PZ ) and P [ ° 3 ; 3] ± [ ° ( logn ) 1 = 3 ; ( logn ) 1 = 3 ]) g ; f n 2 Z + : ( 9 P 2 IPOLY )( n = j PZ \ Z + j and P [ ° 3 ; 3] ± [ ° ( logn ) 1 = 3 ; ( logn ) 1 = 3 ]) g . We also show that if ° 3 ; 3 are replaced by ° 3 = 2 ; 3 = 2 , then both of these sets become recursive. See Corollary 26. 1
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Here IPOLY is the set of all integral polynomials (polynomials of several variables with integer coe¢ cients into < ). For E ± < , PE is the set of all values of P at arguments from E . Here j j is used for cardinality. log is the natural logarithm. Some readers may be uncomfortable with the informal notion of ²mathe- matically natural³ in this context. We can restate the aim as one of ±nding examples that are ²more and more mathematically natural³. Readers who are still uncomfortable will ±nd rich connections between analysis, recursion theory, and Hilbert°s 10th problem of interest in their own right. 2 Some Recursion Theoretic Sets We use N for the set of all nonnegative integers. In this section, we will use W e for the r:e: enumeration of the r:e: sets given by a standard Turing machine model, and e ( n ) for a standard partial recursive enumeration of the partial recursive functions also given by a standard Turing machine model. In section 3, we will work with any standard enumeration.
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