Resultants and discriminants of the multiplication polynomials of Jacobi elliptic functions (Q1711696)

The resultant of two polynomials \[ \begin{aligned} f(x) & =a(x-\alpha_1)\dots(x-\alpha_m), \quad a\neq 0,\\ g(x) & =b(x-\beta_1)\dots(x-\beta_n), \quad b\neq 0,\end{aligned} \] is defined by \[ \mathrm{res}(f,g):=a^nb^m\prod_{i=1}^{m}\prod_{j=1}^{n}(\alpha_i-\beta_j)=a^n\prod_{i=1}^{m}g(\alpha_i). \] For Jacobi's elliptic functions \(x=\mathrm{sn}(u)\), \(y=\mathrm{cn}(u)\), \(z=\mathrm{dn}(u)\), (note that \(y^2=1-x^2\) and \(z^2=1-k^2x^2\)), there exist polynomials \(A_n,B_n,C_n,D_n\) such that (for \(n\) odd) \[ \mathrm{sn}(nu)=\frac{x\,A_n(x)}{D_n(x)},~\mathrm{cn}(nu)=\frac{y\,B_n(x)}{D_n(x)},~\mathrm{dn}(nu)=\frac{z\,C_n(x)}{D_n(x)} \] (a similar identity holds for \(n\) even). The coefficients of \(A_n,B_n,C_n,D_n\) belong to \(\mathbb Z[k^2]\). The main result of the paper is a formula for \(\mathrm{res}(A_n,B_n)\) of the form \[ \mathrm{res}(A_n,B_n)=\kappa_nk^{2l_n}(1-k^2)^{m_n}, \] where \(\kappa_n,l_n,m_n\) are simple terms in \(n\) like, e.g., \(\kappa_n=2^{n^2(n^2-1)/3}\). Analogous identities are given for \(\mathrm{res}(A_n,C_n)\), \(\mathrm{res}(A_n,D_n)\), etc.