##
The recognition problem for equivariant singularities

* Nonlinearity * **1** (1988) 215-240

** Ian Melbourne **

** Abstract **

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.