nature of diophantine solutions in general I had a question in my mind from many years.  we generally present the trivial solutions  to  Diophantine equations. Diophantine equations usually always have some sort of trivial solution (if you allow zero even Fermat’s equation has trivial solutions: $0^n+x^n=x^n$). I suppose the hard question is whether there are non-trivial solutions? The question of non-trivial solutions is already tough for polynomial Diophantine equations and requires detailed knowledge in algebraic and arithmetic geometry, modular forms, etc (see Wiles’ proof of Fermat’s last theorem).

What I am looking for, can we find non-trivial solutions to any polynomial Diophantine equations? If yes, what are the method so far existing?

Thanks in advance to all members of mse.
This question is Hilbert's 10th problem, from Hilbert's famous list of 23 problems promulgated 112 years ago in 1900.

The work of Julia Robinson, Martin Davis, Hillary Putnam, and Yuri Matijasevich, culminating in 1970, showed that it lacks an algorithmic solution.
A polynomial Diophantine equation is an indeterminate polynomial equation whose solutions are restricted to be polynomials in the indeterminate. A Diophantine equation, in general, is one where the solutions are restricted to some algebraic system, typically integers

So you can refer to this wiki article and also this one for a background. Apart from that there is another article that serves a good start. For those who can access IEEE journals, this one is a very nice one.

Thank you and all the best.