Representing logical conjunction of several non-zero conditions as simple expression

If I have two (or more) numbers $x, y \in \mathbb{R}$ and I am interested in the logical conditions $A$ as $x=0$ and $B$ as $y=0$. I can represent logically $A \land B$ as $x^2+y^2=0$. ie $x^2+y^2=0 \iff x=0 \land y=0$. Is there an alternative function with similar properties that ideally also uses basic field operations $+$, $-$, $\times$ and perhaps $\div$?

The condition "$x^2+y^2=0$" is phrased in terms of $\times$ and $+$: namely, it's just "$x\cdot x+ y\cdot y=0$." Even though it uses the language of exponents, that's not necessary.