On polynomials over prime fields taking only two values on the multiplicative group (Q1590973)

Let \(p\) be an odd prime, denote by \(F_p\) the finite field of \(p\) elements, and put \(F_p^*=F_p\setminus\{0\}\). The author proves that if a polynomial over \(F_p\) of degree at most \(p-2\) takes only two values on \(F_p^*\), then one of the following three assertions is true: (1) \(f(X)=a+bX^{(p-1)/2}\), \(a\in F_p\), \(b\in F_p^*\); (2) \(p\equiv 1\bmod 3\) and \(f\) is a polynomial of \(X^{(p-1)/3}\); (3) \(\deg(f)\geq 3(p-1)/4\). Moreover, he disproves the conjecture that the degree of \(f\) has to be of the form \(\frac{d-1}{d}(p-1)\).