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+(φ(x)-z)², where φ(x)∈ R[x] has degree at least 3. Then, for every finite A,B,C⊂ R each of size n, one has |f(A,B,C)|=Ω(n5/3-ε), for every ε>0, where the constant of proportionality depends on ε and on deg(φ). 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 Ω(n3/2).