Transnormal graphs of polynomial curves

We take up the study of transnormal graphs again. We recall that some results on this topic appeared in [2], [3], [4] and [8]. The purpose of this paper is two-fold. We start by considering curves in Euclidean n-space and extend the results in [3]. The main result shows that no polynomial curve of positive even degree (see below for the definitions) has transnormal graph. In section 3 we deal with polynomial functions f: of odd degree. Some conditions are proved to be necessary and sufficient for the transnormality of the graph. The results obtained enable us to characterize the polynomial functions of degree less than or equal to 5 which have transnormal graph.To Professor N. K. Stephanidis on his 65^th birthday