Frobenius non classical curves

Frobenius nonclassicality with respect to linear systems of curves of arbitrary degree

“…Even for this case, however, a complete characterization of F q -Frobenius nonclassical curves is lacking. As observed by Hefez and Voloch [9], characterizing all such curves seems a quite complex problem.…”

“…Frobenius (non)classicality in the case s = 1 has been widely investigated with many examples cited in the literature ( [2], [5], [6], [9]). Even for this case, however, a complete characterization of F q -Frobenius nonclassical curves is lacking.…”

On the number of rational points of a plane algebraic curve

“…To estimate the number of F q -rational points of X , we will use a procedure similar to that in [6]. To do this, we go on to study the ramification divisor R and the F q -Frobenius divisor S of X .…”

“…In [6] the authors pointed out that N = d(q − d + 2) when X is Frobenius non-classical, that is the image Fr(P ) of a generic point P of X under the Frobenius map lies in the tangent line at P . Note that for p > 2 any Frobenius non-classical curve is non-classical in the sense that every point of the curve is an inflexion.…”

Frobenius non-classical curves

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The proof of Proposition 5 of [3] is incomplete. With notation as in the paper, the possibility that the polynomial X~/q'P o + X~/~' P1 + X~/q" Pz in (11) could be identically zero was overlooked.

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