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Concept
Diophantine set
Summary
In mathematics, a Diophantine equation is an equation of the form P(x1, ..., xj, y1, ..., yk) = 0 (usually abbreviated P(, ) = 0) where P(, ) is a polynomial with integer coefficients, where x1, ..., xj indicate parameters and y1, ..., yk indicate unknowns.
A Diophantine set is a subset S of , the set of all j-tuples of natural numbers, so that for some Diophantine equation P(, ) = 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. The use of natural numbers both in S and the existential quantification merely reflects the usual applications in computability and model theory. It does not matter whether natural numbers refer to the set of nonnegative integers or positive integers since the two definitions for Diophantine set are equivalent. We can also 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 is a notoriously hard open problem.
The MRDP theorem (so named for the initials of the four principal contributors to its solution) states that a set of integers is Diophantine if and only if it is computably enumerable. A set of integers S is computably enumerable if and only if there is an algorithm that, when given an integer, halts if that integer is a member of S and runs forever otherwise. 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.
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