# Hessian (algebraic curve)

*of an algebraic curve of degree $n$*

The set of points whose conic polars can be split into two straight lines, as well as the set of double points of the first polars. The Hessian of a non-singular curve of degree $n$ is a curve of degree $3(n-2)$ and class $3(n-2)(3n-7)$. If $f=0$ is the equation of a curve of degree $n$ in homogeneous coordinates and if $f_{ik}=\partial^2f/\partial x_i\partial x_k$, then

$$\begin{vmatrix}f_{11}&f_{12}&f_{13}\\f_{21}&f_{22}&f_{23}\\f_{31}&f_{32}&f_{33}\end{vmatrix}=0$$

is the equation of the Hessian. The Hessian of a non-singular curve of degree 3 in characteristic not equal to three intersects the curve at nine ordinary points of inflection. Named after O. Hesse (1844).

#### Comments

#### References

[a1] | J.L. Coolidge, "A treatise on algebraic plane curves" , Dover, reprint (1959) |

**How to Cite This Entry:**

Hessian (algebraic curve).

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Hessian_(algebraic_curve)&oldid=32348