Sums of distinct polynomial residues (Q6546691)

Let \(f\) be a polynomial with integral coefficients, let \(V_f\) be the value set of \(f\) modulo a prime number \(p\), and let \(S(f)\) denote the sum of all elements of \(V_f\). If \(f(x) = x^2\), then \(S_f\) is the sum of all quadratic residues of \(f\), and we have \(S(f) \equiv 0 \bmod p\). For quadratic polynomials \(f(x) = ax^2 + bx + c\) with \(p \nmid a\), the main result of \textit{S. S. Gross} et al. [Ir. Math. Soc. Bull. 79, 31--37 (2017; Zbl 1430.11042)] is \(S(f) \equiv -\frac{\Delta}{8a} \bmod p\), where \(\Delta = b^2 - 4ac\).\N\NIn this article, the authors compute the sum \(S(f)\) for cubic polynomials.