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LINEAR DIOPHANTINE EQUATIONS (given natural numbers $a, b, c$, are there natural numbers $x$ and $y$ such that $ax + by + c = 0$?) are solvable in polynomial time.

QUADRATIC DIOPHANTINE EQUATIONS ($ax^2 + by + c = 0$) are NP-complete (NP-complete decision problems for quadratic polynomials).

General DIOPHANTINE EQUATIONS are undecidable (Davis-Putnam-Robinson-Matiyasevich theorem).

Are there other classes of Diophantine equations (with restrictions on their arguments/variables) that capture other complexity classes (in particular PSPACE) ?
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This paper, Computational complexities of Diophantine equations with parameters by Tung, proves co-NP-completeness of a variant with parameters over natural numbers.

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Note that it depends a lot on what set are you solving over. For example, the NP-complete SUBSET-SUM problem can be considered as a LINEAR DIOPHANTNE EQUATION, when you restrict your solution over positive integers. If you allow also negative solutions then it is solvable in polynomial time. For an excellent survey, see:

[http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.114.3864][1]

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