Symmetric homotopy construction
Numerical homotopy continuation methods may be applied to compute all of the finite complex solutions to a system of polynomial equations, say \(P(x)= 0\). For systems which are in addition, symmetric, i.e., invariant under a group \(G\) of permutations of the variables and equations, the determination of the zeros can be greatly simplified, since then for each solution \(x\), also \(P(\gamma x)= 0\) for all \(\gamma\in G\), i.e., \(P\) vanishes on the orbit of \(x\) for the group \(G\). In this paper the authors investigate the construction of homotopies which carry the symmetry of the system \(P\). This reduces correspondingly the number of homotopy paths which need to be traversed.
scientific article; zbMATH DE number 643784
scientific article