Hilbert10th1[1].15.96

Hilbert10th1[1].15.96 - 1. Let G be a linear subspace of Q...

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SOME DECISION PROBLEMS RELATED TO HILBERT'S TENTH PROBLEM by Harvey M. Friedman Department of Mathematics Ohio State University January 15, 1996 PRELIMINARY REPORT Let Q n be the n-th Cartesian power of Q = rationals. Q n is viewed at a vector space over Q. An element of Q n is nonzero if and only if it is not the zero element of Q n . Qn can also be viewed as a ring, where a square of x in Q n is the vector x 2 , obtained by squaring each coordinate of x. We will consider linear subspaces of Q n , as well as affine subspaces of Q n . The latter are simply translates by an element of Q n of a linear subspace of Q n . Linear subspaces of Q n are presented by any basis, and affine subspaces of Qn are presented by any basis together with an element of Q n serving as the translate. Obviously there is a decision procedure in a vector and presentation, for determining membership of the vector in the affine subspace with the given presentation. Consider the following decision problems:
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Unformatted text preview: 1. Let G be a linear subspace of Q n . Does G contain the square of a nonzero integral element of G? 2. Let G be a linear subspace of Qn. Does G contain x 2 + 1 for some integral x in G? 3. Let A be an affine subspace of Q n . Does A contain the square of an integral element of A? 4. Let A 1 ,...,A k be affine subspaces of Q n . Do there exist mutually orthogonal integral elements x 1 in A 1 , x 2 in A 2 , . .. , x k in A k ? THEOREM. There exists n such that problems 2 and 3 have no decision procedure. There exists k and n such that problem 4 has no decision procedure. THEOREM. The following are equivalent: i) problem 1 has a decision procedure that works in all dimensions at once; ii) there is a decision procedure for determining whether or not any polynomial with rational coefficients has a rational solution (Hilbert's 10th problem on the rationals)....
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Hilbert10th1[1].15.96 - 1. Let G be a linear subspace of Q...

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