Any Diophantine equation can be reduced to one of at most 11 variables and degree at most around 10^63. No algorithm can decide solvability in rational numbers for this class of Diophantine equations.
That sounds like the coefficients might have to be arbitrarily large. Otherwise all DE's could reduce to a finite set of them, impossible via the MRDP theorem. So it's not so easy to call that bounded complexity.
Any Diophantine equation can be reduced to one of at most 11 variables and degree at most around 10^63. No algorithm can decide solvability in rational numbers for this class of Diophantine equations.