Using symmetries in the eigenvalue method for polynomial systems

One way of solving polynomial systems of equations is by computing a Gröbner basis, setting up an eigenvalue problem and then computing the eigenvalues numerically. This so-called eigenvalue method is an excellent bridge between symbolic and numeric computation, enabling the solution of larger systems than with purely symbolic methods. We investigate the case that the system of polynomial equations has symmetries. For systems with symmetry, some matrices in the eigenvalue method turn out to have special structure. The exploitation of this special structure is the aim of this paper. For theoretical development we make use of SAGBI bases of invariant rings. Examples from applications illustrate our new approach.