Frobenius nonclassical components of curves with separated variables

Abstract

We establish a relation between minimal value set polynomials defined over F q and certain q-Frobenius nonclassical curves. The connection leads to a characterization of the curves of type g(y) = f (x), whose irreducible components are q-Frobenius nonclassical. An immediate consequence will be the realization of rich sources of new q-Frobenius nonclassical curves.

Points on singular Frobenius nonclassical curves

“…One can find several examples of Frobenius nonclassical curves that ilustrate the previous results (see [1] and [2]). Let us consider the particular curve C : x 4 y 2 + x 2 y 4 + x 4 yz + xy 4 z + x 4 z 2 + x 2 y 2 z 2 + y 4 z 2 + x 2 z 4 + xyz 4 + y 2 z 4 = 0 (4.1) over F 4 .…”

On the Dickson–Guralnick–Zieve curve

“…Furthermore, |Γ P | ≥ q 2 (q 2 − 1) and |Γ (1) P | = q 2 . Actually, |Γ P | = q 2 (q 2 − 1) and Γ (2) P is trivial by Result 2. Then (14) reads…”

“…Actually C is isomorphic over F 8 to the well known Klein curve of equation x 3 y + y 3 z + z 3 x = 0. This implies that Aut(C) ∼ = P GL (3,2).…”

Bounds for the number of points on curves over finite fields

“…For the F q 3 -Frobenius nonclassical curve (see [1], [8], [10]) Y q,3 : x q 2 +q+1 + y q 2 +q+1 = 1, it is known that N 1 := #Y q,3 (F q 3 ) = q 5 − q 3 − q 2 + 1 and ǫ 2 = q. The 3(q 2 + q + 1) inflection points P i ∈ F (F q 3 ) have order-sequence (j 0 , j 1 , j 2 ) = (0, 1, q 2 + q + 1), and then B(P i ) ≥ q 2 .…”