Construction of division polynomials I'm trying to understand the construction of the division polynomials used in Schoof's algorithm. I firstly followed this report of Charlap and Robbins.

I stuck with the definition of the leading term p11.

I'm also reading this book, which does the construction also in characteristic 2. However I have the same problem here. 

I'm not able to use the definition in order to computer the leading coefficient of a rational function. However the construction of division polynomials needs me to compute the leading coefficients of the rational function which multiplies a point by m.

Can anybody give me an explicit example of how to compute the leading coefficient?
Leading coefficient of a rational function was defined at 5.3
on page 11 of Report 31
as another function (multiplied by (x/y)^d for some d)
evaluated at identity.

Handling the case of point at identity was explained at top of page 8,
namely multiplying numerator and denominator by y^d and x^d
to make them of the same degree,
resulting in another rational function that is finite and not zero at identity.

Degree of a polynomial of canonical form was defined at 3.6 on page 5;
reduction to this form was explained after definition 3.1 on page 3.

Evaluating a rational function at identity is explained at definition 3.9 page on 5
as the ratio of leading terms of numerator and denominator.

I would expect reduction to canonical form to be a likely reason of confusion.

UPD: For explicit example, let's take a look at Proposition 7.6, leading coefficient of $g_n(x)$ evaluated at identity, with $(g(P), h(P))$ defined at the start of Section 7 "Multiplication by $n$" and $g(P)$ depending only on $x$-coordinate of the point (Exercise 7.5). It was proved by induction on page 23. Corollary 7.7 is likely of major interest for such a question.