Hilbert's Tenth Problem

A Resolution to Hilberts First Problem

The continuum hypothesis (CH) is one of and if not the most important open problems in set theory, one that is important for both mathematical and philosophical reasons. The general problem is determining whether there is an infinite set of real numbers that cannot be put into one-to-one correspondence with the natural numbers or be put into one-to-one correspondence with the real numbers respe...

The Hilbert’s-Tenth-Problem Operator

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 view HTP as an operator, mapping each set W of prime numbers to HTP (Z[W−1]), which is naturally viewed as a set of polynomials in Z[X1, X2, . . .]. For W = ∅, it is a famous result of Matiyasevich, Davis, Putnam, and Robinson that the jump ∅′ is Turing-equivale...

Extensions of Hilbert's Tenth Problem

Lectures Two types of lectures were programmed into the schedule. First there were a number of introductory lectures which were were meant to introduce non-specialist into the area of Hilbert’s Tenth Problem. Since quite a few of the participants came from different backgrounds this was an ideal way to get a feeling of the problems which come up in this area of mathematics. The second type of l...

Quantum Algorithm for the Hilbert’s Tenth Problem

We propose a quantum algorithm for the classically non-computable Hilbert’s tenth problem, which ultimately links to the Turing halting problem. Quantum continuous variables and quantum adiabatic evolution are employed for an implementation. Also discussed are a method for the time estimation for the adiabatic evolution, and a comparison with more the well-known quantum computation employing a ...

Notes on Hilbert’s Tenth Problem over Q