Up to this point we have concentrated on the algebraic side of the description of plane curve singularities. It is even more fascinating to try and visualise them. After a preliminary section on vector fields, in which we recall some standard results from analysis which will give us our main tool for constructing homeomorphisms, we go on to a detailed geometrical description of the local behaviour of a curve at a singular point, which gives in particular a picture of the topology of the link.

We go on to calculate the numerical invariants needed to specify the particular knot or link. Using some basic results about the Alexander polynomial of a knot leads to our main conclusion, that the topology determines the numerical invariants defined earlier.

Vector fields

In this section we develop our main technique for constructing diffeo-morphisms. A first idea is to start with a diffeomorphism somewhere and deform it in a 1-parameter family. We thus define a smooth isotopy from X to Y to be a smooth embedding F : X × I → Y × I of the form F(x, t) = (ft(x), t), so that each ft is a smooth embedding of X into Y ; we also say that the embeddings f0 and f1 are isotopic. The fundamental case is when Y = X and we start at the identity map f0(x) = x.

The second idea is to differentiate F with respect to the ‘time’ variable t. For each P ∈ X, ft(P) describes a smooth curve in X, which thus has a tangent vector at each point.

In the preceding Chapters, we have encountered several sets of real numbers of Lebesgue measure zero, including the set of Liouville numbers, the set of real numbers with bounded partial quotients, the set of very well approximable numbers, and the set of S*-numbers of *-type strictly greater than 1. Some of them are certainly strictly larger than others: indeed, as it may be seen by considering continued fraction expansions (see Exercise 1.5), there are very well approximable numbers other than the Liouville numbers. On the other hand, the set of S*-numbers of *-type at least 2 contains the set of S*-numbers of *-type at least 3, but the results of Chapters 1 to 4 do not enable us to decide whether the inclusion is strict or not.

In the present Chapter, we introduce a powerful tool for discriminating between the sets of Lebesgue measure zero, namely the notion of Hausdorff dimension, developed by Hausdorff in 1919 [276]. Shortly thereafter, Jarník [288, 292] and, independently, Besicovitch [100], applied it to number theoretical problems, and they determined the Hausdorff dimension of sets of real numbers very close to infinitely many rational numbers (Theorem 5.2). Their result has been subsequently generalized in many directions. For instance, A. Baker and Schmidt [45] showed in 1970 that there exist S*-numbers of arbitrarily large but finite *-type (Theorem 5.5). In the present Chapter, we prove both these results and we quote some other extensions of the Jarník–Besicovitch Theorem. Further refinements are stated in Chapter 6.

The theorem of Puiseux states that a polynomial equation f(x, y) = 0 has a solution in which y is expressed as a power series in fractional powers of x. In this chapter we will give several versions of this theorem, of increasing sharpness. In the first section we present the classical algorithm for calculating the successive terms in the power series, and show that this does yield a solution. However, to obtain a convergent power series requires more work, and in the second section we give a different approach giving an introduction to the geometry of the situation and an existence proof for convergent power series solutions.

The next short section collects the results describing the relations between curves, their branches, tangents and multiplicities, which are basic for later chapters.

The fourth section establishes some basic properties of the rings of power series, in particular that they are unique factorisation domains, and deduces that the solutions obtained in the preceding sections must all be the same.

Solution in power series

We want to solve a polynomial equation f(x, y) = 0. There are several ways to find a solution for y in terms of x, but we begin with one which gives an effective method of calculation. For this, it will make no difference if we allow f to be a formal power series. The basis of the method of proof goes back to Newton [142].

Central to the study of the topology attached to the Milnor fibration is understanding the monodromy. As well as the monodromy map on homology we consider the Seifert form. These two, together with the intersection form, form a single algebraic structure which gives a rather fine invariant of the topology, and enables a number of numerical invariants to be picked out. We will use the decomposition of the Milnor fibre obtained in the preceding chapter: this permits simplified proofs of a number of basic results.

A Seifert form can be defined for any knot or link provided with a spanning Seifert surface, but in the case of fibred knots such a surface is canonically provided, so that the Seifert form is intrinsic in this case. Both Seifert forms and monodromy can be defined and studied in a higher dimensional situation, but there the canonical decomposition is lacking, so proofs are more sophisticated.

The chapter opens by defining the Seifert form and eliciting its algebraic properties. Next we derive the special features of Seifert forms for the case of fibred knots using the JSJ decomposition established in Chapter 9. Using the model constructed in Section 9.3 we obtain an analysis of algebraic properties of the monodromy.

We then investigate Seifert forms in the abstract in sufficient detail to obtain in principle enough invariants to classify them with rational coefficients. This section involves algebraic technicalities, and the reader may choose to omit it since the Seifert forms arising for curve singularities have special properties.

In Theorem 3.3, we used the Borel–Cantelli Lemma 1.2 to prove that almost all real numbers ξ are S*-numbers of *-type less than or equal to 1. A similar statement is however much more difficult to establish when we consider Mahler's classification. The first important result in this direction is due to Mahler [377], who showed in 1932 that almost all real numbers ξ satisfy supn≥1 (wn (ξ)/ n) ≤ 4. At the end of [377], he made the conjecture that the upper bound 4 could be replaced by 1.

Until Sprindžuk [536, 537, 538] gave in 1965 a complete affirmative answer to that conjecture, there appeared various improvements of Mahler's result. First, Koksma [333] showed that supn≥1 (wn (ξ) / n) ≤ 3 for almost all real numbers ξ. This has been strengthened by LeVeque [362], who replaced the upper bound 3 in Koksma's result by 2. Later on, refining a method introduced by Kasch and Volkmann [311], Schmidt [501] proved the inequality wn (ξ) ≤ 2n – 7/3 for almost all real numbers ξ and all positive integers n. Lastly, Volkmann [583, 584] showed that wn (ξ) ≤ 4n/3 holds for almost all real numbers ξ and all positive integers n. At the same time, Sprindžuk [533] obtained a slightly stronger result than Volkmann's, shortly before his resolution of Mahler's Conjecture.

In Chapter 8 we took a geometric approach to the combinatorics associated to a curve singularity, studying functions on the resolution tree. In this chapter we give a more algebraic presentation. This gives interesting information about the set of ideals in the local ring O0: = Ox, y of O. We obtain a relation between these ideals and ‘clusters’ of infinitely near points, which can be formulated as a Galois correspondence between these.

This has two applications. One is a procedure (Enriques' ‘unloading algorithm’) leading from a numerical definition of an ideal to the effective numerical parameters defining it. The other is a lead in to the study of integral closures of ideals: we establish the surprisingly close connection between integrally closed ideals and exceptional cycles.

We briefly address the question of determinacy, that is, finding for each reduced f ∈ O0 the least integer n such that the terms of degree n in the power series expansion of f are sufficient to determine the equisingularity type of the curve Cf.

In the final section we briefly discuss properties of plane curve singularities from the viewpoint of the local ring OC, which is that taken in modern algebraic geometry.

Blowing up ideals

We study ideals I in the ring O0, (which can be identified with ℂ{x, y}) of germs at O of holomorphic functions on the plane T0. We begin by showing how I gives rise to an ideal in the local ring of the surface obtained by blowing up at a point, and establishing some basic results relating I to these blown up ideals.

Throughout the present Chapter, we are essentially concerned with the following problem: for which functions Ψ : ℝ≥1 → : ℝ≥0 is it true that, for a given real number ξ, or for all real numbers ξ in a given class, the equation |ξ – p/q| < Ψ (q) has infinitely many solutions in rational numbers p/q? We begin by stating the results on rational approximation obtained by Dirichlet and Liouville in the middle of the nineteenth century. In Section 1.2, we define the continued fraction algorithm and recall the main properties of continued fractions expansions. These are used in Section 1.3 to give a full proof of a metric theorem of Khintchine. The next two Sections are devoted to the Duffin–Schaeffer Conjecture and to some complementary results on continued fractions.

Dirichlet and Liouville

Every real number ξ can be expressed in infinitely many ways as the limit of a sequence of rational numbers. Furthermore, for any positive integer b, there exists an integer a with |ξ – a/b| ≤ 1/(2b), and one may hope that there are infinitely many integers b for which |ξ – a/b| is in fact much smaller than 1/(2b). For instance, this is true when ξ is irrational, as follows from the theory of continued fractions.

Geometry of numbers turns out to be a very useful tool in Diophantine approximation. For instance, it allows us to construct non-zero integer polynomials taking small values at prescribed points. In the course of the book, we applied several times the ‘first Theorem of Minkowski’ and the ‘second Theorem of Minkowski’, which are Theorems B.2 and B.3 below, respectively. We give a full proof of Theorem B.2, but not of Theorem B.3, which is much deeper. Throughout this Appendix. n denotes a positive integer. A set C in ℝn having inner points and contained in the closure of its open kernel is called a body (or a domain).

theorem B.1. Let C be a bounded convex body in ℝn, symmetric about the origin and of volume vol(C). If vol(C) > 2nor if vol(C) = 2nand C is compact, then C contains a point with integer coordinates, other than the origin.

proof. This proof is due to Mordell [429]. By classical arguments from elementary topology, it is enough to treat the case where vol(C) > 2n. For any positive integer m, denote by Cm the set of points of C having rational coordinates with denominator m. As m tends to infinity, the cardinality of Cm becomes equivalent to vol(C)mn, and is thus strictly larger than (2m)n when m is large enough.