🤖 AI Summary
This work proposes a novel and efficient method for computing isolated regular solutions of multivariate polynomial systems with composable structure. By introducing variable substitutions, the original system is reduced to a lower-dimensional system in intermediate variables. The approach integrates a probabilistic symbolic homotopy algorithm, algebraic independence analysis, and the Chevalley–Shephard–Todd theorem to handle cases invariant under finite reflection groups. It is the first systematic exploitation of composability in polynomial systems to substantially reduce symbolic solving complexity. The algorithm achieves arithmetic complexity polynomial in both the input size and the number of solutions. Experimental results on benchmark systems—including those invariant under symmetric groups, hyperoctahedral groups, and exceptional reflection groups—demonstrate its superior computational efficiency.
📝 Abstract
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 \(E_6, E_7, E_8, F_4, H_3, H_4\).