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- Diophantine_set abstract "In mathematics, a Diophantine equation is an equation of the form P(x1, ..., xj, y1, ..., yk)=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 of integers S is recursively 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. 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.".
- Diophantine_set wikiPageExternalLink 12d.pdf.
- Diophantine_set wikiPageExternalLink DiophantineSet.html.
- Diophantine_set wikiPageExternalLink Matiyasevich_theorem.
- Diophantine_set wikiPageID "101700".
- Diophantine_set wikiPageRevisionID "605917316".
- Diophantine_set hasPhotoCollection Diophantine_set.
- Diophantine_set subject Category:Diophantine_equations.
- Diophantine_set subject Category:Hilbert's_problems.
- Diophantine_set type Abstraction100002137.
- Diophantine_set type Attribute100024264.
- Diophantine_set type Communication100033020.
- Diophantine_set type Condition113920835.
- Diophantine_set type Difficulty114408086.
- Diophantine_set type DiophantineEquations.
- Diophantine_set type Equation106669864.
- Diophantine_set type Hilbert'sProblems.
- Diophantine_set type MathematicalStatement106732169.
- Diophantine_set type Message106598915.
- Diophantine_set type Problem114410605.
- Diophantine_set type State100024720.
- Diophantine_set type Statement106722453.
- Diophantine_set comment "In mathematics, a Diophantine equation is an equation of the form P(x1, ..., xj, y1, ..., yk)=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.".
- Diophantine_set label "Diophantine set".
- Diophantine_set sameAs m.0psqx.
- Diophantine_set sameAs Q16251580.
- Diophantine_set sameAs Q16251580.
- Diophantine_set sameAs Diophantine_set.
- Diophantine_set wasDerivedFrom Diophantine_set?oldid=605917316.
- Diophantine_set isPrimaryTopicOf Diophantine_set.