{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Hilbert10th1[1].15.96

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

This preview shows pages 1–2. Sign up to view the full content.

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:

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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)....
View Full Document

{[ snackBarMessage ]}