Toric geometry and equivariant bifurcations

Many problems in equivariant bifurcation theory involve the computation of invariant functions and equivariant mappings for the action of a torus group. We discuss general methods for finding these based on some elementary considerations related to toric geometry, a powerful technique in algebraic geometry. This approach leads to interesting combinatorial questions about cones in lattices, which lead to explicit calculations of minimal generating sets of invariants, from which the equivariants are easily deduced. We also describe the computation of Hilbert series for torus invariants and equivariants within the same combinatorial framework. As an example, we apply these methods to the interaction of two linear modes of a Euclidean-invariant PDE on a rectangular domain with periodic boundary conditions.