1. Classification of singularities

In a tremendous work, V. I. Arnold started, in the late sixties, the classification of hypersurface singularities up to right equivalence. Here $f$ and $g \in K\langle x_1, \dots, x_n\rangle$ are called right equivalent if they coincide up to analytic coordinate transformation, that is, if there exists a local $K$-algebra automorphism $\varphi$ of $K\langle x \rangle$ such that $f = \varphi(g)$. 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).

$$\begin{array}{cclc} A_k & : & x_1^{k+1} + x_2^2 + x_3^2 + \... ... E_8 & : & x_1^5 + x_2^3 + x_3^2 + \dots + x_n^2. & \end{array}$$

Arnold introduced the concept of ``modality'', related to Riemann's idea of moduli, into singularity theory and classified all singularities of modality $\le 2$ (and also of Milnor number $\le 16$). The ADE singularities are just the singularities of modality 0. Singularities of modality 1 are the three parabolic singularities:

$$\begin{array}{lcll} \widetilde{E}_6 = P_8 = T_{333} & : & x^... ...36} & : & x^3 + y^5 + ax^2y^2, & 4a^3 + 27 \not= 0, \end{array}$$

the 3-indexed series of hyperbolic singularities

$$T_{pqr} : x^p + y^q + z^r + axyz, a \not= 0, \dfrac{1}{p} + \dfrac{1}{q} + \dfrac{1}{r} < 1$$

and 14 exceptional families, cf. AGV.

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 $f$, there exists an integer $k$ such that $f$ and $g$ are right equivalent if their Taylor expansion coincides up to order $k$. Therefore, knowing the determinacy $k$ of $f$, we can replace $f$ by its Taylor polynomial up to order $k$.

The determinacy can be estimated as the minimal $k$ such that

$${\mathfrak{m}}^{k+1} \subset {\mathfrak{m}}^2$$    jacob$$(f)$$

where ${\mathfrak{m}} \subset K\langle x_1, \dots, x_n\rangle$ is the maximal ideal and jacob $(f) = \langle \partial f/\partial x_1, \dots, \partial f/\partial x_n\rangle$. Hence, this $k$ can be computed by computing a standard basis of ${\mathfrak{m}}^2$ jacob$(f)$ and normal forms of ${\mathfrak{m}}^i$ with respect to this standard basis for increasing $i$, using a local monomial ordering. However, there is a much faster way to compute the determinacy directly from a standard basis of ${\mathfrak{m}}^2$    jacob$(f)$, which is basically the ``highest corner'' described in GP1.

An important initial step in Arnold's classification is the generalised Morse lemma, or splitting lemma, which says that $f \circ \varphi(x_1, \dots, x_n) = x_1^2 + \dots + x_r^2 + g(x_{r+1}, \dots, x_n)$ for some analytic coordinate change $\varphi$ and some power series $g \in {\mathfrak{m}}^3$ if the rank of the Hessian matrix of $f$ at 0 is $r$.

The determinacy allows the computation of $\varphi$ up to sufficiently high order and a polynomial $g$ 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 $T_{5,7,11}$ from a database A$_-$L (``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:

1. standard bases for local and global orderings;
2. computation of invariants (Milnor number, determinacy, $\ldots$);
3. generalised Morse lemma;
4. syzygies for local orderings.

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.