Several complex variables and analytic spaces Complex singularities Conjecture

Zariski's multiplicity conjecture

Builds on Ring of convergent power series Germ Homeomorphism

Statement

Let $f, g \in \C\{z_0, \dots, z_n\}$ be convergent power series without repeated factors, with $f(0) = g(0) = 0$ . The multiplicity $\operatorname{mult}_0 f$ is the least degree of a nonzero homogeneous term in the power series expansion of $f$ . Suppose that there is a homeomorphism $\phi$ between neighbourhoods of $0$ in $\C^{n+1}$ with $\phi(0) = 0$ . Suppose that $\phi$ maps the germ of $f^{-1}(0)$ at $0$ onto the germ of $g^{-1}(0)$ at $0$ . Then $\operatorname{mult}_0 f = \operatorname{mult}_0 g$ .

Remarks

Zariski asked this question in his 1971 paper "Some open questions in the theory of singularities". It is open in general. It holds for plane curves, $n = 1$ , where the embedded topological type determines the Puiseux characteristics of the branches and hence the multiplicity. The multiplicity of $f$ equals the multiplicity of the local ring $\C\{z_0, \dots, z_n\} / (f)$ , so the conjecture asks whether this algebraic invariant is determined by the embedded topology. The Milnor number, by contrast, is known to be a topological invariant in this sense.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Germ concept
Homeomorphism concept
Ring of convergent power series concept
Zariski's multiplicity conjecture conjecture

Tag	`ZTTZVZWMK9`
URI	`m2://ZTTZVZWMK9`
Kind	conjecture
Writing	W0 stub
MSC	32S05, 32S15, 14B05
Status	This copy of the site has no build data. Run `m2 status ZTTZVZWMK9`.
About	Singularity theory, Multiplicity of a local ring
Uses	Ring of convergent power series, Germ, Homeomorphism

Zariski's multiplicity conjecture

Statement

Remarks

Reading path

Further reading

What this page needs for W1

What links here