My question is whether there is a polynomial in two variables which, in a precise sense, encodes the ring structure of $\mathbb{C}$:
Is there a two-variable complex polynomial $p(x,y)$ such that complex addition and multiplication are each first-order definable in the structure $(\mathbb{C}; p)$?
This is related to this question of Gregory Nisbet. I earlier asked the three-variable version of this question ... because counting is hard and I didn't realize that a three-variable function corresponds to a four-ary relation. This was the question I meant to ask.
Consider the polynomial $p(x,y)=x^2+y$. We can define $0$ as the unique $a$ such that $p(a,y)=y$ for all $y$. We can then define squaring as $x^2=p(x,0)$, and then define addition as $x+y=p(z,y)$ where $z$ is any element such that $z^2=x$. Once we have addition and squaring, we can define multiplication since $xy$ is the unique element such that $xy+xy=(x+y)^2-x^2-y^2$.
This construction works more generally in any reduced commutative ring in which every element has a square root and $2$ is not a zero divisor. I imagine some of these assumptions can be removed or modified by tweaking the construction, but I suspect a different idea would be needed to get a polynomial that works without some very strong assumption similar in spirit to "every element has a square root".