Computation of the decomposition group of a triangular ideal (Q1762559)

The authors consider the following problem: Given an ideal \(I\) generated by a triangular set \(\{f_1(x_1), f_2(x_1,x_2),\dots, f_n(x_1,\dots,x_n)\}\) in \(k[x_1,\dots,x_n]\), \(k\) a field, compute a strong generating set for the decomposition group \(\text{Dec}(I)=\{\sigma\in S_n: I^\sigma=I\}\). Interesting examples of such systems occur in the computation of splitting fields of polynomials: \(f_1(x)\) has roots \(\{\alpha_1,\dots,\alpha_m\}\) in the algebraic closure \(K/k\) and \(f_r(x,\alpha_{i_2},\dots,\alpha_{i_r})\) are split factors of \(f_1(x)\) over some algebraic extensions. If \(\text{Dec}(I)\) acts transitively and \(I\) is radical this is the general form of ideals under consideration. For the zero set \(V=V(I)\) we have the Bezout bound \(| V| \leq \prod_i{\deg(f_i)}\). \(I\) is called a pure Galois ideal if equality holds and \(V= \text{Dec}(I)\cdot\alpha\) is a single orbit of an element \(\alpha\in V\). The authors describe, for \(G= \text{Dec}(I)\), an algorithm that computes recursively strong generating sets for the stabilizers \(G_k,\;k=n,\dots,0\), that fix \(\{1,\dots,k\}\). In general, due to backtracking the algorithm has exponential complexity. For pure Galois ideals backtracking does not occur and \(\text{Dec}(I)\) can be computed with \(O(n^3)\) normal form computations. The authors report about test computings using a Magma implementation of their algorithms. Note that proposition\,(4.2) is wrong in the stated generality since for \(I=\{f(x_1), x_2-x_1,\dots, x_n-x_1\}\) the group \(\text{Dec}(I)=S_n\) does not act faithfully on \(V\). One has additionally to assume 4.3.(1).