Is there a general formula to compute the number of integer solutions of an equation?

Question

recently, I asked a question concerning the number of solutions of a diophantine equation that used the rounding function. This question, however, dealt with a linear function, and I was wondering if the method or the answer could be generalized to include larger families of functions. I was trying to use the same technique given to me in the answer of that question to solve: $$ \lceil x(\ln (x \ln x))\rceil+ \lceil y(\ln (y \ln y))\rceil = N$$ However, I do not think the same method can be applied, given that this equation is non linear. I have tried but I got stuck in the spot with the minimums and maximums. Is there a function $f(N)$ that counts how many integer solutions this equation has?
Furthermore, is there a function $f(g(x),m,N)$ that counts the number of integer solutions of the following equation? $$\sum_{i=1}^m g(x_i)=N$$

Answer 1

Of course there is such a function: you just defined it. The question is whether there is an algorithm for computing this function.

In general there is no way to tell whether a polynomial Diophantine equation (in several variables) has any solutions: see Hilbert's 10th Problem.

In your case, because the left side is greater than $x + y$ for $x, y \ge 3$, any solution must have $x + y \le N$, so there are only finitely many possibilities to try.