Tjurina number

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.