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Around Sziklai’s conjecture on the number of points of a plane curve over a finite field. (English) Zbl 1194.14031
The paper considers the problem of bounding $$M_q(d)$$, the maximal number of rational points on a plane projective curve of degree $$d$$ over the finite field with $$q$$ elements that does not contain a line. It shows that the question is interesting only when $$2\leq d\leq q+1$$. A conjecture by Sziklai states that $$M_q(d)\leq (d-1)q+1$$, and it is known that $$M_q(d)\leq (d-1)q+\lfloor d/2\rfloor$$. The authors prove that $$M_q(d)\leq (d-1)q+(q+2-d)$$, which is for many $$(d,q)$$ sharper than the best bound known before, and implies Sziklai’s conjecture for $$d=q+1$$. The proof uses a very nice ‘two ways of counting’ the elements of a certain set. The paper also disproves the general conjecture by showing that $$M_4(4)=14$$, a bound which is attained by essentially just one curve.

MSC:
 14G15 Finite ground fields in algebraic geometry 14G05 Rational points 14H25 Arithmetic ground fields for curves
Keywords:
many points; plane curves; bound
manYPoints
Full Text:
References:
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