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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.

##### MSC:
 11G20 Curves over finite and local fields