A note on diagonal and Hermitian hypersurfaces.

Aspects of the properties, enumeration and construction of points on diagonal and Hermitian hypersurfaces have been considered extensively in the literature and are further considered here. The zeta function of diagonal hypersurfaces is given as a direct result of the work of Wolfmann. Recursive construction techniques for the set of rational points of Hermitian hypersurfaces are of interest. The relationship of these techniques here to the construction of codes on hypersurfaces is briefly noted.