To appear in J. Symbolic Computation Zeros of equivariant vector elds: Algorithms for an invariant approach

We present a symbolic algorithm to solve for the zeros of a polynomial vector eld equivariant with respect to a nite subgroup of O(n). We prove that the module of equivariant polynomial maps for a nite matrix group is Cohen-Macaulay and give an algorithm to compute a fundamental basis. Equivariant normal forms are easily computed from this basis. We use this basis to transform the problem of nding the zeros of an equivariant map to the problem of nding zeros of a set of invariant polynomials. Solving for the values of fundamental polynomial invariants at the zeros e ectively reduces each group orbit of solutions to a single point. Our emphasis is on a computationally e ective algorithm and we present our techniques applied to two examples.