Expansion of trivariate polynomials using proximity

We extend the proximity technique of Solymosi and Zahl [J. Combin. Theory, Ser. A (2024)] to the setting of trivariate polynomials. In particular, we prove the following result: Let $f(x,y,z)=(x-y)^2+(\varphi(x)-z)^2$, where $\varphi(x)\in \mathbb{R}[x]$ has degree at least 3. Then, for every finite $A,B,C\subset \mathbb{R}$ each of size $n$, one has $|f(A,B,C)|=\Omega(n^{5/3-\varepsilon})$, for every $\varepsilon>0$, where the constant of proportionality depends on $\varepsilon$ and on ${\rm deg}(\varphi)$. This improves the previous exponent $3/2$, due to Raz, Sharir, and De Zeeuw [Israel J. Math. (2018)]. To the best of our knowledge, prior to this work no trivariate polynomial was known to have expansion exceeding $\Omega(n^{3/2})$.