Improved Bound for the k-Variate Elekes–Rónyai Theorem

Let f ∈ ℝ[x1, . . ., x_k], for k ≥ 2. For any finite sets A1, . . ., A_k ⊂ ℝ, consider the set f(A1, . . ., A_k):= {f(a1, . . ., a_k) | (a1, · · ·, a_k) ∈ A1 × · · · × A_k}, that is, the image of A1 × · · · × A_k under f. Extending a theorem of Elekes and Rónyai, which deals with the case k = 2, and the result of Raz, Sharir, and De Zeeuw [9], dealing with the case k = 3, it is proved in Raz and Shem Tov [10], that for every choice of finite A1, . . ., A_k ⊂ ℝ, each of size n, one has |f(A1, . . ., A_k)| = Ω(n³/²), unless f has some degenerate special form. In this paper, we introduce the notion of a rank of a k-variate polynomial f, denoted as rank(f). Letting r = rank(f), we prove that |f(A1, . . ., A_k)| = Ω (n^{5r−4/2r −ε)} , for every ε > 0, where the constant of proportionality depends on ε and on deg(f). This improves the lower bound (1), for polynomials f for which rank(f) ≥ 3.