We prove the following statement. Let f ∈ ℝ[x1,…,xd], for some d ≥ 3, and assume that f depends non-trivially in each of x1,…, xd. Then one of the following holds.(i)For every finite sets A1,…, Ad ⊂ℝ, each of size n, we have (Formula presented.) with constant of proportionality that depends on deg f.(ii)f is of one of the forms (Formula presented.) or (Formula presented.) for some univariate real polynomials h(x), pi(x),…,pd(x). This generalizes the results from [2,5,7], which treat the cases d = 2 and d = 3.
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