Diophantine Set

Diophantine Set

In mathematics, a Diophantine equation is an equation of the form P(x₁, ..., x_j, y₁, ..., y_k)=0 (usually abbreviated P(x,y)=0 ) where P(x,y) is a polynomial with integer coefficients. A Diophantine set is a subset S of Nj so that for some Diophantine equation P(x,y)=0.

That is, a parameter value is in the Diophantine set S if and only if the associated Diophantine equation is satisfiable under that parameter value. Note that the use of natural numbers both in S and the existential quantification merely reflects the usual applications in computability and model theory. We can equally well speak of Diophantine sets of integers and freely replace quantification over natural numbers with quantification over the integers. Also it is sufficient to assume P is a polynomial over and multiply P by the appropriate denominators to yield integer coefficients. However, whether quantification over rationals can also be substituted for quantification over the integers it is a notoriously hard open problem.

The MRDP theorem states that a set of integers is Diophantine if and only if it is computably enumerable. A set S is recursively enumerable precisely if there is an algorithm that, when given an integer, eventually halts if that input is a member of S and otherwise runs forever. This means that the concept of general Diophantine set, apparently belonging to number theory, can be taken rather in logical or recursion-theoretic terms. This is far from obvious, however, and represented the culmination of some decades of work.

Matiyasevich's completion of the MRDP theorem settled Hilbert's tenth problem. Hilbert's tenth problem was to find a general algorithm which can decide whether a given Diophantine equation has a solution among the integers. While Hilbert's tenth problem is not a formal mathematical statement as such the nearly universal acceptance of the (philosophical) identification of a decision algorithm with a total computable predicate allows us to use the MRDP theorem to conclude the tenth problem is unsolvable.