On the classification of low-degree ovoids of Q(4,q)

Ovoids of the non-degenerate quadric Q(4,q) of PG(4,q) have been studied since the end of the '80s. They are rare objects and, beside the classical example given by an elliptic quadric, only three classes are known for q odd, one class for $q$ even, and a sporadic example for $\`iq=3^5. It is well known that to any ovoid of Q(4,q) a bivariate polynomial f(x,y) can be associated. In this paper we classify ovoids of Q(4,q) whose corresponding polynomial f(x,y) has 'low degree' compared with q, in particular deg(f)<(q/6.3)^(3/13)-1. Finally, as an application, {two classes} of permutation polynomials in characteristic 3 are obtained.