Forms of low degree in finite fields

Abstract

It is known that diagonal forms over a finite field have non-trivial zeros if the number of variables, or the size of the field, is large enough. We consider diagonal forms of degree up to five in cases where the size hypotheses are not satisfied. There are finitely many fields not covered by the known results, but a direct computational test of all possible equations is impractical. We describe means of cutting down considerably on the number of fields and the number of equations for which there exist diagonal forms, of degree up to 5 and in 3 variables, with no non-trivial zero.