On diagonal equations over finite fields

Let Fq be a finite field with q=pn elements. In this paper, we study the number of solutions equations form a1x1d1+…+asxsds=b xi?Fpti, where ai,b?Fq and ti|n for all i=1,…,s. our main results, employ results on quadratic forms to give an explicit formula diagonal restricted solution sets satisfying certain natural restrictions exponents. As consequence, present conditions existence solutions. second part focus case t1=…=ts=n. A classic well-known result from Weil yields bound such d1=…=ds, necessary sufficient equation being maximal minimal respect Weil's bound. particular, completely characterize Fermat type curves. We also discuss further questions concerning some open problems.