Measure theory and Hilbert’s Tenth Problem inside Q
نویسنده
چکیده
For a ring R, Hilbert’s Tenth Problem HTP(R) is the set of polynomial equations over R, in several variables, with solutions in R. When R = Z, it is known that the jump Z′ is Turing-reducible to HTP(Z). We consider computability of HTP(R) for subrings R of the rationals. Applying measure theory to these subrings, which naturally form a measure space, relates their sets HTP(R) to the set HTP(Q), whose decidability remains an open question. We raise the question of the measure of the topological boundary of the solution set of a polynomial within this space, and show that if these boundaries all have measure 0, then for each individual oracle Turing machine Φ, the reduction R′ = ΦHTP(R) fails on a set of subrings R of positive measure. That is, no Turing reduction of the jump R′ of a subring R to HTP(R) holds uniformly on a set of measure 1.
منابع مشابه
Baire category theory and Hilbert’s Tenth Problem inside Q
For a ring R, Hilbert’s Tenth Problem HTP(R) is the set of polynomial equations over R, in several variables, with solutions in R. We consider computability of this set for subrings R of the rationals. Applying Baire category theory to these subrings, which naturally form a topological space, relates their sets HTP(R) to the set HTP(Q), whose decidability remains an open question. The main resu...
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