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The uniqueness of a plane curve of degree q attaining Sziklai's bound over F-qopen access
- Authors
- Homma, Masaaki; Kim, Seon Jeong
- Issue Date
- May-2012
- Publisher
- ACADEMIC PRESS INC ELSEVIER SCIENCE
- Keywords
- Plane curve; Finite field; Rational point
- Citation
- FINITE FIELDS AND THEIR APPLICATIONS, v.18, no.3, pp 567 - 580
- Pages
- 14
- Indexed
- SCI
SCIE
SCOPUS
- Journal Title
- FINITE FIELDS AND THEIR APPLICATIONS
- Volume
- 18
- Number
- 3
- Start Page
- 567
- End Page
- 580
- ISSN
- 1071-5797
1090-2465
- Abstract
- We prove the uniqueness of a plane curve of degree q over a finite field F-q which attains Sziklai's bound q(q - 1) + 1. More precisely, if a plane curve of degree q over F-q has q(q 1) + 1 rational points, then it is projectively equivalent to the curve defined by the equation X-q - XZ(q-1) + Xq-1Y - Y-q = 0. Although the case q = 4 is the exception to Sziklai's bound, the uniqueness of a curve of degree 4 with 13 points over F-4 still holds. (C) 2011 Elsevier Inc. All rights reserved.
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