Reducing complex arithmetic to a three-variable polynomial

Question

This was not the question I wanted to ask originally - I've asked the right version here.

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This is a very small special case of this question of Gregory Nisbet. Say that a polynomial $p$ with complex coefficients captures the complex field iff addition and multiplication of complex numbers are first-order definable in the structure $(\mathbb{C};p)$. For example, the polynomial $$q(x,y,z,w)=(x+y)z-w$$ captures the complex field:

• We can define $0$ as the unique $t$ such that for all $x,y$ we have $q(x,y,t,t)=t$.

• We can then define multiplication as $ab=q(a,0,b,0)$.

• We can define $1$ as the unique $s$ such that for all $x$ we have $q(x,0,s,0)=x$.

• Finally, this lets us define addition as $a+b=q(a,b,1,0)$.

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Following Gregory Nisbet's above-linked question, I'm curious if we can do better, at least as far as the number of variables is concerned:

Is there a polynomial in $3$ variables which captures the complex field?

I've removed some embarrassingly silly guesswork; see the edit history if curious.

Answer 1

You almost already have a solution: Just ignore the $w$ input to your polynomial.

That is, set $f(x,y,z) = (x+y)z$, and define

• $0$ is the unique $t$ such that for all $x, y$ we have $f(x,y,t)=t$.
• $1$ is the unique $u$ such that for all $x$ we have $f(x,0,u)=x$.
• $x+y$ is $f(x,y,1)$.
• $xy$ is $f(x,0,y)$.