Diophantine equations and complexity classes 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) ?
This paper, Computational complexities of Diophantine equations with parameters by Tung, proves co-NP-completeness of a variant with parameters over natural numbers.