30,518 research outputs found
On the Dickson-Guralnick-Zieve curve
The Dickson-Guralnick-Zieve curve, briefly DGZ curve, defined over the finite
field arises naturally from the classical Dickson invariant of
the projective linear group . The DGZ curve is an
(absolutely irreducible, singular) plane curve of degree and genus
In this paper we show that the DGZ curve has
several remarkable features, those appearing most interesting are: the DGZ
curve has a large automorphism group compared to its genus albeit its
Hasse-Witt invariant is positive; the Fermat curve of degree is a
quotient curve of the DGZ curve; among the plane curves with the same degree
and genus of the DGZ curve and defined over , the DGZ curve
is optimal with respect the number of its -rational points
Elliptic curves and explicit enumeration of irreducible polynomials with two coefficients prescribed
Let be a finite field of characteristic . We give the number of
irreducible polynomials x^m+a_{m-1}x^{m-1}+...+a_0\in\F_q[x] with
and prescribed for any given if , and with and
prescribed for if .Comment: 17 pages, Part of the results was presented at the Polynomials over
Finite Fields and Applications workshop at Banff International Research
Station, Canad
- …