Stepanov method of the estimation of the number of roots of some equations (Q1316875)

Let \(p\) be an odd prime, and \(\mathbb{F}_ p\) a finite field with \(p\) elements. \(L(x)= \sum_{k=1}^{p-1} x^{p-k}/k\) and \(E(x)= \sum_{k=0}^{p-1} x^ k/k!\) are two polynomials over \(\mathbb{F}_ p\). In the present paper, the author proves that for any \(\lambda\in \mathbb{F}_ p\) the number of roots of \(L(x)-\lambda\) or \(E(x)-\lambda\) in \(\mathbb{F}_ p\) does not exceed \(2p^{2/3}+2\).