an:06125843
Zbl 1259.14023
Homma, Masaaki; Kim, Seon Jeong
Nonsingular plane filling curves of minimum degree over a finite field and their automorphism groups: Supplements to a work of Tallini
EN
Linear Algebra Appl. 438, No. 3, 969-985 (2013).
00312357
2013
j
14G15 14H37 14H50 14G05 11G20
plane curve; finite field; rational point; automorphism group of a curve
Let \(\mathbb{F}_q\) be a finite field. The authors find nonsingular projective plane curves over \(\mathbb{F}_q\), of degree \(q+2\), whose \(\mathbb{F}_q\)-rational points fill the whole projective plane \(\mathbb{P}^2(\mathbb{F}_q)\). The degree \(q+2\) is minimal for a curve with this property. Let \(x,y,z\) be homogeneous coordinates of \(\mathbb{P}^2\), and \(U=y^qz-yz^q\), \(V=z^qx-zx^q\), \(W=x^qy-xy^q\). The curves have equations \(F_A=0\), where \(F_A=(x\;y\;z)A(U\;V\;W)^t\), for a \(3\times 3\) matrix \(A\) with entries in \(\mathbb{F}_q\), whose characteristic polynomial is irreducible. The automorphism groups of these curves are also determined.
The results of the paper are similar to those obtained by \textit{G. Tallini} in [Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 30, 706--712 (1961; Zbl 0107.38104), Rend. Mat. Appl., V. Ser. 20, 431--479 (1961; Zbl 0106.35604)].
Enric Nart Vi??als (Barcelona)
Zbl 0107.38104; Zbl 0106.35604