# Lower degree terms of elliptic curve division polynomials

Suppose that I have $$E$$ which is an elliptic curve over finite field $$\mathbb F_{23}$$ expressed by the equation: $$E: y^2=x^3+7$$

The $$n^{th}$$ division polynomials $$\psi_n$$ are defined with:$$\psi_1=1$$ $$\psi_2=2y$$ $$\psi_3=3x^4+6ax^2+12bx-a^2$$ $$\psi_4=4y(x^6+5ax^4+20bx^3-5a^2x^2-4abx-8b^2-a^3)$$ $$\psi_{2m+1}=\psi_{m+2}\psi^3_m\psi_{m-1}\psi^3_{m+1}$$ $$\psi_{2m}=(2y)^{-1}(\psi_{m+2}\psi^2_{m-1}-\psi_{m-2}\psi^2_{m+1})\psi_m$$

In some papers there is a definition:

$$\psi_n=y(nx^{(n^2-4)/2})$$+(lower degree terms) for $$n$$ is even and:

$$\psi_n=nx^{(n^2-1)/2}$$+(lower degree terms) for $$n$$ is odd

Division polynomial is multivariate one(integers a,b,x,y is "input"). So all division polynomials can be rewritten to:$$\sum_{i_1,i_2,i_3,i_4=0}^n=a_{i_1i_2i_3i_4}x^{i_1}y^{i_2}A^{i_3}B^{i_4}$$

(from the definition of the polynomial)

My question is whether it is possible to derive equation for lower $$n$$-th degree terms just like equation for leading term written above to have full polynomial equation? If no, why leading term can be derived and lower terms cannot?

As far as I know all division polynomials form an polynomial ring. According to this paper (Lemma 2.1.5) for $$n$$ is odd $$\psi_n$$ is in $$\mathbb Z[A,B,x,y^2]$$ (in case of this finite field $$\mathbb F_{23}[A,B,x,y^2]$$) and for $$n$$ is even $$(2y)^{-1}\psi_n$$ is in $$\mathbb F_{23}[A,B,x,y^2]$$. Since $$y^2=x^3+Ax+B$$ the ring is in the form of $$\mathbb F_{23}[A,B,x]$$. So I had an idea to manage it from that ring which have coefficients to $$\psi_n$$ since $$\mathbb F_{23}[A,B,x]$$ is isomorphic to $$\mathbb F_{23}[A][B][x]$$ but I have trouble understanding it. And polynomial ring was my first idea for initial approach to the problem so my reasoning may be wrong.

Yes, you can do as you say, and write all odd division polynomials $$\psi_n$$ (or $$(2y)^{-1}\psi_n$$ for even $$n$$) for $$y^2=x^3+Ax+B$$ (or a long Weierstrass equation if needed) as polynomials in the ring $$\mathbb{Z}[A,B,x]$$. However, the polynomials are defined recursively, so I have never seen a closed formula for the $$n$$-th division polynomial. The highest-order term can be derived from the theory of formal groups for instance, but the lower order terms are messy... For example, here is $$\psi_5(x)$$: $$5x^{12} + 62Ax^{10} + 380Bx^9 - 105A^2x^8 + 240ABx^7 + (-300A^3 - 240B^2)x^6 - 696A^2Bx^5 + (-125A^4 - 1920AB^2)x^4 + (-80A^3B - 1600B^3)x^3 + (-50A^5 - 240A^2B^2)x^2 + (-100A^4B - 640AB^3)x + A^6 - 32A^3B^2 - 256B^4.$$ You can use a computer to compute any given division polynomial in $$\mathbb{Z}[A,B,x]$$. For example, I used Magma to compute the one above, using the following code:
F<A,B>:=FunctionField(Rationals(),2);