A note on value sets of quartic polynomials

Let v   be the number of distinct values of the polynomial f(x)=x4+ax2+bxf(x)=x4+ax2+bx, where a and b are elements of the finite field of size q, where q is odd. When b is 0, an exact formula for v can be given. When b   is not 0, v=(5/8)q+O(q), where the error term comes from the Riemann hypothesis. In this note we establish for the case that b   is not 0, the inequality v≥(q+1)/2v≥(q+1)/2, without relying on the Riemann hypothesis.