Expansion of Trivariate Polynomials Using Proximity

We extend the proximity technique of Solymosi and Zahl [9] 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 [7]. To the best of our knowledge, prior to this work no trivariate polynomial was known to have expansion exceeding Ω(n³/²).