Is there an elegant way to find the singularities of an algebraic variety

[frb]

Let V be the variety of the ideal (f)

a singular point is a point where all the partial derivatives of the f are zero.
I know you can find singular points by writing down all these partial derivatives and also that the points are zeros of f (such as all points on the variety) and solve that system of equations. These are generally very difficult systems to solve so I wondered if there was a more elegant method to find these singular points.

 

[Chris Hillman]

Have you read Cox, Little, O'Shea, Ideals, Varieties and Algorithms? One of the best books ever published.

 

[frb]

No, but I was trying to avoid resorting to mere number crunching. Seems like there is no other way...

 

[Chris Hillman]

I was thinking of computational algebra as in "symbolic computation", not numerical computation. Groebner bases are a student's best friend!

 

[mathwonk]

if by "find" you mean compute, why does "number crunching" seem inappropriate?