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Polynomials modulo \(p\) whose values are squares (elementary improvements on some consequences of Weil’s bounds). (English) Zbl 1064.11501
From the text: In this paper the author gives an elementary proof that the curve \(Y^2=f(X)\) has at least \(\sqrt{2p}-\frac32-d\) solutions in \(\mathbb F_p^2\) with \(Y\neq0\) where \(f(X)\) is a polynomial of degree \(d\) over \(\mathbb F_p\) with simple roots.. This is much weaker than the Weil bound, however it shows the existence of solutions if \(p\) is sufficiently large compared to \(d\). The author’s proof uses repeated application of a certain differential operator and comparison of degrees. For the special case \(d=3\) (i.e., an elliptic curve), the author gives a very short proof.

11G20 Curves over finite and local fields