So suppose we have a functional inequality with polynomial arguments in $2$ variables, $\sum_i c_i f(p_i(x,y)) \geq 0$, where $c_i$ are say given integer constants and $p_i$ are given polynomials, say with integer coefficients. We want the inequality to hold for all $x,y \in \mathbb{R}$.
Is it a decidable problem to determine whether there are non-constant real-valued functional solutions for $f:{\mathbb R} \to {\mathbb R}$?
A recent problem of this flavor came up on math.stackexchange, which was to determine any non-constant solutions to $f(x) + f(y) \geq f(x + y) + f(xy)$. I wondered whether it was possible to determine the existence question with an algorithm rather than ad hoc analysis.