A Symbolic Homotopy Algorithm for Solving Composable Polynomial Systems

We study the problem of computing the isolated regular solutions of a system $(f_1,\ldots,f_n)$ of $n$ polynomial equations in $n$ variables $(X_1, \dots, X_n)$ over a field of characteristic zero $k$. We focus on systems with a \emph{composable structure}, where each polynomial $f_i$ can be expressed as a composition $f_i = h_i(g_1,\dots,g_n)$. Exploiting this structure allows us to reduce the original system to one in the $g_j$ variables, thereby significantly improving the efficiency of symbolic solution algorithms. We present a probabilistic algorithm that computes all isolated regular solutions, with arithmetic complexity being polynomial in the input size and in the number of solutions. A first important application is when $f_1, \dots, f_n$ belong to the subring $k[g_1, \dots, g_n]$, where $g_1, \dots, g_n$ are algebraically independent polynomials in $k[X_1, \dots, X_n]$. Another important application is to systems of invariant polynomials under finite reflection groups, since by the Chevalley-Shephard-Todd theorem their invariant rings are polynomial algebras. Typical examples include the symmetric groups $S_n$, the hyperoctahedral groups $B_n$, the dihedral groups $I_2(m)$, and the exceptional finite reflection groups $E_6, E_7, E_8, F_4, H_3, H_4$.