Let $E$ be an elliptic curve defined over the real numbers $R$ and let $P \in E(R)$. In this note we give an elementary proof of necessary and sufficient conditions for the preimages of $P$ under duplication to be real-valued.
"Real Preimages of Duplication on Elliptic Curves." Missouri J. Math. Sci. 29 (1) 19 - 26, May 2017. https://doi.org/10.35834/mjms/1488423698