# Is deciding whether there is a non-constant solution to a functional inequality with polynomial arguments decidable, with 2 variables?

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.

• Where do you want the inequality to hold? Do you want it to hold for all $x,y$? To make sure I understand, the $p_i,c_i$ are given as inputs, and the goal is to decide whether there exists a non-constant function $f$ that makes the inequality hold for all $x,y$? Are there any requirements on $f$, such as that it must be continuous or differentiable? – D.W. May 1 '14 at 21:58
• @D.W. no constraints on $f$, other than satisfying the inequality, but yes the inequality must hold for all $x,y$. And the input is $p_i,c_i$ like you said. – user2566092 May 1 '14 at 22:03
• OK, I edited your question to make this clearer. In the future, you can edit your question by clicking "edit" underneath it, and we like you to edit the question to make it self-contained so people don't have to read the comment thread to understand the question. – D.W. May 1 '14 at 22:14