Primitive free cubics with specified norm and trace

Abstract

Abstract. The existence of a primitive free (normal) cubic x 3 − ax 2 + cx − b over a finite field F with arbitrary specified values of a ( = 0) and b (primitive) is guaranteed. This is the most delicate case of a general existence theorem whose proof is thereby completed.

Frobenius nonclassical components of curves with separated variables

“…Remark 5.3. Recall that a detailed description of W in the case k = 2 is given in (14). From that, it can be verified that all irreducible curves y q+1 = f (x) arising from Theorem 5.6 are F q 2 -isomorphic to the Hermitian curve.…”

“…Part of the motivation to determine N k−1 (u, v) is given by its known connections with many other problems (e.g. [14], [18], [24]). Accordingly, this new relation with certain Frobenius nonclassical curves establishes an additional motivation.…”

On Katz's bound for the number of elements with given trace and norm

“…This bound was used by Moisio [3] to improve some cases of the explicit bound in Wan [5] on the number of irreducible polynomials in an arithmetic progression of F q [x]. In the case n + 1 = 3, the Katz bound also plays a significant role in Cohen and Huczynska [1] for their proof of the existence of a cubic primitive normal polynomial with given norm and trace.…”

“…The second step is to use the detailed cohomological calculations in Rojas-Leon and Wan [4] for such toric hypersurfaces. In the case n + 1 = 3, the above improved bounds should significantly reduce the amount of calculations in [1].…”

Kloosterman sums, elliptic curves, and irreducible polynomials with prescribed trace and norm

“…proved by using deep algebraic geometry. The Katz bound with m = 3 plays a significant role in the proof by Cohen and Huczynska [10] of the existence of a primitive free (normal) cubic polynomial with a ( = 0) and b fixed, which completed a general existence theorem (see also [3,5,4]). We shall improve the Katz bound in this case.…”