Is there a P-complete problem on diophantine equations? In general deciding whether a diophantine equation has any integer solutions is equivalent to the halting problem.  I believe that deciding if a quadratic diophantine equation has any solution is NP-complete. Does there exist a further restriction on the equations involved that yields a P-complete problem?
No , as far as i know diaphantine problem in general is undecidable thus equivalent to halting problem , if the equations are restricted to be quadratic then it is np-complete , and linear diaphantine equation can be reduced to integer programming problem and For linear Diophantine equation equations, integral solutions exist if and only if, the GCD of coefficients of the two variables divides the constant term perfectly.