= Division polynomials
{title2=$\psi_n$}
{wiki}
= Division polynomial of an elliptic curve
{synonym}
= Division polynomials of an elliptic curve
{synonym}
For an <elliptic curve> $E:y^2=x^3+Ax+B$ in <characteristic> different from two and three, division <polynomials> are elements of its <coordinate ring> encoding multiplication and <torsion points of an elliptic curve>. The initial terms include
$$
\psi_1=1,\qquad\psi_2=2y,\qquad\psi_3=3x^4+6Ax^2+12Bx-A^2.
$$
The recurrences for higher $n$ yield
$$
x([n]P)=x(P)-\frac{\psi_{n+1}(P)\psi_{n-1}(P)}{\psi_n(P)^2}.
$$
For odd $n$, $\psi_n$ is a <polynomial> in $x$; for even $n$, it has a factor $2y$, and $\psi_n/(2y)$ is a <polynomial> in $x$. Thus the phrase “roots of the division <polynomial>” must specify whether it means <points> on the <curve> or roots in the $x$-line. When $n$ is prime to the <characteristic>, its zero <points> are exactly the nonidentity $n$-<torsion points of an elliptic curve>; for odd $n$, the two opposite <points> share an $x$-root.<Isogeny kernels and division polynomials by Andrew Sutherland>[Pages 10–11, initial terms, Lemma 5.20 and Theorem 5.21]{ref}
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