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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
Software:
manYPoints
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References:
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