I have two families of polynomials $\{\psi_n \}_{n \in \mathbb{N}}$ and $\{ h_m \}_{m \in \mathbb{N}} $ in $\mathbb{Z}[y, l]$. The polynomials are pretty nice. Each $\psi_n$ is quadratic in $y$,
$$\psi_n = \sum_{j=1}^{2n-1} (a_{n, j} + b_{n,j}y^2)l^j $$
and $h_m = \sum_{j=0}^{m-1} c_{m, j} l^j \in \mathbb{Z}[l]$. Both families can be defined recursively.
In a prefect world, I would like to be able to compute an explicit or recursive formula for the resultant of $\psi_n$ and $h_m$ with respect to $l$ (eliminating $l$ from the polynomial system $\psi_m = 0 = h_n$), but this seems like a lot to ask.
Instead, is there a way to compute the leading coefficient or term of $\mathrm{Res}_l ( \psi_n, h_m)$?
Reference to papers etc. would be much appreciated.