On a theorem of Morlaye and Joly and its generalization

Abstract: We show that a weaker version of the well-known theorem of Morlaye and Joly on diagonal equations is a simple consequence of a restricted variable version of the Chevalley-Warning theorem. Moreover, we extend the result of Morlaye and Joly to the case of an equation of the form $$b_1D_{m_1}(X_1,a_1)+\dots+b_nD_{m_n}(X_n,a_n)=c,$$ where $D_{m_1}(X_1,a_1),\dots,D_{m_n}(X_n,a_n)$ are Dickson polynomials.