---
tag: RF66FF634A
kind: definition
name: Tjurina number
msc: [32S05, 14B07, 32S30]
defines: SWX539TCG0
uses: [0NAHXR1FRH, 096Z9MP2ME, VZXF7RNFKS]
sources:
  - url: https://en.wikipedia.org/wiki/Milnor_number
---

## Definition

Let $\mathcal{O}_n = \C\{x_1, \ldots, x_n\}$ be the ring of [[germs|VZXF7RNFKS]] at $0$ of [[holomorphic functions|0NAHXR1FRH]] on $\C^n$, that is, convergent power series.
Let $f \in \mathcal{O}_n$ with $f(0) = 0$, and let $I$ be the [[ideal|096Z9MP2ME]] generated by $f$ and its partial derivatives.
The *Tjurina number* of $f$ is
$$\tau(f) = \dim_\C \mathcal{O}_n / I, \qquad I = \left( f, \frac{\partial f}{\partial x_1}, \ldots, \frac{\partial f}{\partial x_n} \right).$$
It may be infinite.

## Remarks

The quotient is the Tjurina algebra of $f$.
The number $\tau(f)$ is finite if and only if the hypersurface $f = 0$ has an isolated singular point at $0$ or is smooth there.
Example: for $f = x^2 + y^3$ the ideal is $(x^2 + y^3, 2x, 3y^2) = (x, y^2)$, so $\mathcal{O}_2 / (x, y^2)$ has basis $1, y$ and $\tau(f) = 2$.
Non-example: for $f = x^2$ on $\C^2$ the ideal is $(x)$, and $\mathcal{O}_2 / (x) \cong \C\{y\}$ is infinite-dimensional; the hypersurface $x^2 = 0$ is singular along the whole line $x = 0$, so the singularity is not isolated.
The Milnor number is $\mu(f) = \dim_\C \mathcal{O}_n / (\partial f / \partial x_1, \ldots, \partial f / \partial x_n)$, so $\tau(f) \le \mu(f)$.
For an isolated singularity, $\tau(f) = \mu(f)$ if and only if $f$ is quasi-homogeneous in suitable coordinates, by a theorem of K. Saito.
The Tjurina number depends only on the hypersurface germ, not on the choice of $f$: if $u$ is a unit of $\mathcal{O}_n$, then $uf$ and $f$ give the same ideal.