In a tremendous work, V. I. Arnold started, in the late sixties, the classification of hypersurface singularities up to right equivalence. Here and are called right equivalent if they coincide up to analytic coordinate transformation, that is, if there exists a local -algebra automorphism of such that . His work culminated in impressive lists of normal forms of singularities and, moreover, in a determinator for singularities which allows the determination of the normal form for a given power series ([AGV, II.16]). This work of Arnold has found numerous applications in various areas of mathematics, including singularity theory, algebraic geometry, differential geometry, differential equations, Lie group theory and theoretical physics. The work of Arnold was continued by C.T.C. Wall and others, cf. Wa,GKr.
Most prominent is the list of ADE or simple or Kleinian singularities, which have appeared in surprisingly different areas of mathematics, and still today, new connections of these singularities to other areas are being discovered (cf. Gre2 for a survey). Here is the list of ADE singularities (the names come from their relation to the simple Lie groups of type A, D and E).
Arnold introduced the concept of ``modality'', related to Riemann's idea of moduli, into singularity theory and classified all singularities of modality (and also of Milnor number ). The ADE singularities are just the singularities of modality 0. Singularities of modality 1 are the three parabolic singularities:
The proof of Arnold for his determinator is, to a great part constructive, and has been partly implemented in SINGULAR, cf. Kr. Although the whole theory and the proofs deal with power series, everything can be reduced to polynomial computation since we deal with isolated singularities, which are finitely determined. That is, for an isolated singularity , there exists an integer such that and are right equivalent if their Taylor expansion coincides up to order . Therefore, knowing the determinacy of , we can replace by its Taylor polynomial up to order .
The determinacy can be estimated as the minimal such that
An important initial step in Arnold's classification is the generalised Morse lemma, or splitting lemma, which says that for some analytic coordinate change and some power series if the rank of the Hessian matrix of at 0 is .
The determinacy allows the computation of up to sufficiently high order and a polynomial as in the theorem. This has been implemented in SINGULAR and is a cornerstone in classifying hypersurface singularities.
In the following example we use SINGULAR to get the singularity from a database AL (``Arnold's list''), make some coordinate change and determine then the normal form of the complicated polynomial after coordinate change.
LIB "classify.lib"; ring r = 0,(x,y,z),ds; poly f = A_L("T[5,7,11]"); f; ==> xyz+x5+y7+z11 map phi = r, x+z,y-y2,z-x; poly g = phi(f); g; ==> -x2y+yz2+x2y2-y2z2+x5+5x4z+10x3z2+10x2z3+5xz4+z5+y7-7y8+21y9-35y10 ==> -x11+35y11+11x10z-55x9z2+165x8z3-330x7z4+462x6z5-462x5z6+330x4z7 ==> -165x3z8+55x2z9-11xz10+z11-21y12+7y13-y14 classify(g); ==> The singularity ... is R-equivalent to T[p,q,r]=T[5,7,11]
Ingredients for the classification of singularities:
Beyond classification by normal forms, the construction of moduli spaces for
singularities, for varieties or for vector bundles is a pretentious goal,
theoretically as well as computational. First steps towards this goal for
singularities have been undertaken in Ba and FrK.