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The uniqueness of a plane curve of degree $$q$$ attaining Sziklai’s bound over $$\mathbb F_{q}$$. (English) Zbl 1243.14024
Let $$C$$ be a projective plane curve of degree $$d$$ defined over the finite field $$k$$ with $$q$$ elements. The curve $$C$$ may be reducible but with no $$k$$-linear components. Let $$N_q(C)$$ denote the number of $$k$$-rational points of $$C$$. P. Sziklai [Finite Fields Appl. 14, No. 1, 41–43 (2008; Zbl 1185.14017)] conjectured that $$N_q(C)\leq (d-1)q+1\, (*)$$ except for the curve over $${\mathbb F}_4$$ $$X^4+Y^4+Z^4+X^2Y^2+Y^2Z^2+Z^2X^2+X^2YZ+XY^2Z+XYZ^2=0$$ which has 14 $$F_4$$-rational points. This conjecture was proved by M. Homma and S. J. Kim [Finite Fields Appl. 16, No. 5, 315–319 (2010; Zbl 1196.14030)]. In the paper under review the authors are interested in those curves of degree $$d=q$$ attaining equality in $$(*)$$. In fact this problem has already been considered by the authors for $$d=q+1$$ [Finite Fields Appl. 17, No. 3, 240–253 (2011; Zbl 1215.14033)] and $$d\geq q+2$$ [“Nonsingular plane filling curves of minimum degree over a finite field and their automorphism groups: Supplements to a work of Tallini”, arXiv:0903.1918]. There is just one curve, up to projective equivalence, of degree $$q$$ namely $$X^{q}-XZ^{q-1}+X^{q-1}Y-Y^q=0$$ whose number of $${\mathbb F}_q$$-rational points equals $$(q-1)q+1$$.

##### MSC:
 14H50 Plane and space curves 14G15 Finite ground fields in algebraic geometry 14G05 Rational points 14N10 Enumerative problems (combinatorial problems) in algebraic geometry
##### Keywords:
plane curve; finite field; rational point
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##### References:
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