Assume that we want to find all integer (or rational) solutions to the polynomial Diophantine equation $$ P(x_1,\dots,x_n) = 0 $$ where $P$ is a polynomial with integer coefficients. Do we have a standard definition/convention what exactly do we mean by "solving" such equation? If the number of solutions is finite this is clear: we need to list all the solutions and prove that there is no more. But what if the solution set is infinite? What description of this infinite set counts as "the solution" of the equation?
A trivial example is the equation $xy=0$, whose (integer) solution set is $(x,y)=(0,u)$ or $(x,y)=(u,0)$, where $u$ is an integer parameter. More generally, a polynomial family is the set of the form $(P_1(y_1,\dots,y_k),\dots,P_n(y_1,\dots,y_k))$, where $P_i$ are polynomials with integer coefficients, and $y_j$ are integer parameters. If we have represented the solution set of an equation as a finite union of polynomial families, then we certainly solved the equation. Unfortunately, for many equations, for example Pell equation $$ x^2 - 2y^2 = 1 $$ such representation of solution set is impossible. However, for this equation, it is easy to describe the solution set using recurrence relations.
Another standard example is finding rational solutions to equations that defines an elliptic curve. In this case the solution set forms a finitely generated group, and the set of generators of this group counts as a complete solution.
How to unify these examples in general definition? In each case, we have a complete description of the solution set, and this description can easily be used to enumerate/list all solutions. However, if we define that equation is solved if there is an algorithm that lists all its solutions, then all equations are trivially "solved" by an algorithm that just tries all candidate solutions in some order.
I am looking for a formal definition/convention in the form "We say that an equation is solved if and only if ... ".
As a specific example, what does it mean to find all integer solutions to the trivially-looking equations such as $$ x^2 + x = yz $$ or $$ x^2 + x + 1 = yz ? $$ We can define the divisor function $D(n)$ that, given a positive integer $n$, returns all its divisors. Then the solution set to the second equation is $(x,y,z)=(u, v, (u^2+u+1)/v)$, where $u \in {\mathbb Z}$ and $v \in D(u^2+u+1)$. But does this count as an adequate description of the solution set? Can we use the divisor function $D$? What exactly we can or cannot use?
