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

In summary, a singular point is a point where all partial derivatives of a function (f) are equal to zero. These points can be found by solving a system of equations, which is often a difficult task. However, there are more elegant methods available such as using computational algebra, specifically Groebner bases, to find these points. This approach is preferred over traditional number crunching methods.
  • #1
frb
16
0
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.
 
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  • #2
Have you read Cox, Little, O'Shea, Ideals, Varieties and Algorithms? One of the best books ever published.
 
  • #3
No, but I was trying to avoid resorting to mere number crunching. Seems like there is no other way...
 
  • #4
I was thinking of computational algebra as in "symbolic computation", not numerical computation. Groebner bases are a student's best friend!
 
  • #5
if by "find" you mean compute, why does "number crunching" seem inappropriate?
 

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

What is an algebraic variety?

An algebraic variety is a set of points in n-dimensional space that satisfy a system of polynomial equations. It can also be described as the zero set of a polynomial function in n variables.

What are singularities of an algebraic variety?

Singularities are points on an algebraic variety where the polynomial equations defining the variety are not well-defined or do not have a unique solution. They can be thought of as points where the variety is "broken" or has unusual behavior.

Why is it important to find the singularities of an algebraic variety?

Finding the singularities of an algebraic variety is important for understanding the geometry and structure of the variety. Singularities can provide valuable information about the behavior of the variety and can also help in solving systems of polynomial equations.

What is an elegant way to find the singularities of an algebraic variety?

An elegant way to find the singularities of an algebraic variety is by using the method of elimination theory. This involves eliminating variables from the system of polynomial equations until a system of equations in one variable is obtained. The solutions of this system correspond to the singular points of the original variety.

Are there any other methods for finding singularities of an algebraic variety?

Yes, there are various other methods for finding singularities of an algebraic variety, such as the use of Gröbner bases, resultants, and intersection theory. However, the method of elimination theory is often considered the most elegant and efficient method for finding singularities.

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