The set of division polynomials is a sequence of polynomials in Z [ x , y , A , B ] {\displaystyle \mathbb {Z} [x,y,A,B]} with x , y , A , B {\displaystyle x,y,A,B} free variables that is recursively defined by:
The polynomial ψ n {\displaystyle \psi _{n}} is called the nth division polynomial.
Using the relation between ψ 2 m {\displaystyle \psi _{2m}} and ψ 2 m + 1 {\displaystyle \psi _{2m+1}} , along with the equation of the curve, the functions ψ n 2 {\displaystyle \psi _{n}^{2}} , ψ 2 n y , ψ 2 n + 1 {\displaystyle {\frac {\psi _{2n}}{y}},\psi _{2n+1}} , ϕ n {\displaystyle \phi _{n}} are all in K [ x ] {\displaystyle K[x]} .
Let p > 3 {\displaystyle p>3} be prime and let E : y 2 = x 3 + A x + B {\displaystyle E:y^{2}=x^{3}+Ax+B} be an elliptic curve over the finite field F p {\displaystyle \mathbb {F} _{p}} , i.e., A , B ∈ F p {\displaystyle A,B\in \mathbb {F} _{p}} . The ℓ {\displaystyle \ell } -torsion group of E {\displaystyle E} over F ¯ p {\displaystyle {\bar {\mathbb {F} }}_{p}} is isomorphic to Z / ℓ × Z / ℓ {\displaystyle \mathbb {Z} /\ell \times \mathbb {Z} /\ell } if ℓ ≠ p {\displaystyle \ell \neq p} , and to Z / ℓ {\displaystyle \mathbb {Z} /\ell } or { 0 } {\displaystyle \{0\}} if ℓ = p {\displaystyle \ell =p} . Hence the degree of ψ ℓ {\displaystyle \psi _{\ell }} is equal to either 1 2 ( l 2 − 1 ) {\displaystyle {\frac {1}{2}}(l^{2}-1)} , 1 2 ( l − 1 ) {\displaystyle {\frac {1}{2}}(l-1)} , or 0.
René Schoof observed that working modulo the ℓ {\displaystyle \ell } th division polynomial allows one to work with all ℓ {\displaystyle \ell } -torsion points simultaneously. This is heavily used in Schoof's algorithm for counting points on elliptic curves.