The recognition problem for equivariant singularities

Nonlinearity 1 (1988) 215-240

Ian Melbourne


Singularity theory involves the classification of singularities up to some equivalence relation. The solution to a particular recognition problem is the characterisation of an equivalence class in terms of a finite number of polynomial equalities and inequalities to be satisfied by the Taylor coefficients of a singularity.

The recognition problem can be simplified by decomposing the group of equivalences into a unipotent group and a group of matrices. Building upon results of Bruce et al , we show for contact equivalence that in many cases the unipotent problem can be solved by just using linear algebra. We give a necessary and sufficient condition for this, namely that the tangent space be invariant under unipotent equivalence. We then develop efficient methods for checking whether the tangent space is invariant, and give several examples drawn from equivariant bifurcation theory.