---
tag: QMMSKHEW9S
kind: statement
role: theorem
name: Mather–Yau theorem
msc: [32S05, 32S25, 14B05]
uses: [VTGYEES3VS, EVPZ7ZG1HB, 096Z9MP2ME]
about: [VTGYEES3VS]
sources:
  - url: https://en.wikipedia.org/wiki/Milnor_number
---

## Statement

Let $\mathcal{O}_{n+1} = \C\{z_0, \dots, z_n\}$ be the [[ring of convergent power series|EVPZ7ZG1HB]].
For $f \in \mathcal{O}_{n+1}$ with $f(0) = 0$, let the *Tjurina algebra* be
$$A(f) = \mathcal{O}_{n+1} \Big/ \left( f, \frac{\partial f}{\partial z_0}, \dots, \frac{\partial f}{\partial z_n} \right),$$
the quotient by the [[ideal|096Z9MP2ME]] generated by $f$ and its partial derivatives.
Let $f, g \in \mathcal{O}_{n+1}$ have [[isolated singularities|VTGYEES3VS]] at $0$.
Then the following are equivalent:

1. the hypersurface germs $(f^{-1}(0), 0)$ and $(g^{-1}(0), 0)$ are biholomorphic, that is, there are a biholomorphic germ $\phi$ of $(\C^{n+1}, 0)$ and a unit $u \in \mathcal{O}_{n+1}$ with $g = u \cdot (f \circ \phi)$;
2. $A(f)$ and $A(g)$ are isomorphic as $\C$-algebras.

## Remarks

John Mather and Stephen Yau proved the theorem in 1982.
The direction from 1 to 2 is a direct computation, since the ideal $(f, \partial f / \partial z_j)$ is carried to $(g, \partial g / \partial z_j)$ by $\phi$ and $u$.
The converse is the hard part.
Its proof uses the finite determinacy of isolated singularities and a study of the action of the group of coordinate changes on jets.

Example: for $f = x^2 + y^{k+1}$ with $k \ge 1$, the ideal is $(x, y^k)$, so $A(f) \cong \C[y] / (y^k)$.
These algebras have different dimensions for different $k$, so the singularities $A_k$ are pairwise not biholomorphic.

The dimension $\tau(f) = \dim_\C A(f)$ is the *Tjurina number*.
It satisfies $\tau(f) \le \mu(f)$, and K. Saito proved that $\tau(f) = \mu(f)$ if and only if $f$ is right equivalent to a weighted homogeneous polynomial.

## Explanations

### L1 human

Two hypersurfaces with an isolated singular point may look different only because of the choice of coordinates.
The theorem gives a finite test for whether they are really the same: compute one finite-dimensional algebra for each and compare.
So a geometric classification problem becomes a problem about finite-dimensional commutative algebras.

### L2 human

The Tjurina algebra is also the space of first-order deformations of the hypersurface germ: its dimension $\tau$ is the number of parameters of the miniversal deformation.
The theorem says that this algebra remembers the whole singularity.
Together with the [[Milnor number|1ZE51P7GRQ]] and the [[classification of simple singularities|YE5F548WMS]], it is a basic tool of the analytic classification of isolated hypersurface singularities.