Intersection of irreducible curves and the Hermitian curve

Let H(q )denote the Hermitian curve in P-2 over Fq(2) and C-d be an irreducible plane projective curve in P-2 also defined over Fq(2) of degree d. Can Hq and Cd intersect in exactly d(q+1) distinct Fq(2) -rational points? B & eacute;zout's theorem immediately implies that Hq and Cd intersect in at most d(q+1) points, but equality is not guaranteed over Fq(2) . In this paper we prove that for many d <= q(2)-q+1, the answer to this question is affirmative. The case d=1 is trivial: it is well known that any secant line of H-q defined over Fq(2) intersects Hq in q+1 rational points. Moreover, all possible intersections of conics and Hq were classified in [9] and their results imply that the answer to the question above is affirmative for d=2 and q >= 4, as well. However, an exhaustive computer search quickly reveals that for (q,d)is an element of{(2,2),(3,2),(2,3)}, the answer is instead negative. We show that for q <= d <= q2-q+1, d=& LeftFloor;(q+1)/2 & RightFloor; and d=3, q >= 3 the answer is again affirmative. Various partial results for the case d small compared to q are also provided.