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

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Discussion Overview

The discussion revolves around the methods for finding singularities of an algebraic variety defined by an ideal. Participants explore the challenges of identifying singular points, particularly focusing on the computational aspects and the potential for more elegant solutions beyond traditional methods.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant defines a singular point as where all partial derivatives of the defining polynomial are zero and acknowledges the difficulty of solving the resulting system of equations.
  • Another participant suggests consulting the book "Ideals, Varieties and Algorithms" by Cox, Little, and O'Shea as a valuable resource.
  • A participant expresses a desire to avoid "number crunching," implying a preference for more elegant or theoretical approaches to the problem.
  • One participant introduces the concept of computational algebra and symbolic computation, highlighting the usefulness of Groebner bases in this context.
  • A later reply questions the appropriateness of dismissing "number crunching" when it comes to computing singular points.

Areas of Agreement / Disagreement

Participants express differing views on the methods for finding singularities, with some advocating for computational techniques while others seek more theoretical approaches. The discussion remains unresolved regarding the best method to employ.

Contextual Notes

Participants have not reached consensus on the most effective approach to finding singularities, and there are varying interpretations of what constitutes an "elegant" method.

frb
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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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Have you read Cox, Little, O'Shea, Ideals, Varieties and Algorithms? One of the best books ever published.
 
No, but I was trying to avoid resorting to mere number crunching. Seems like there is no other way...
 
I was thinking of computational algebra as in "symbolic computation", not numerical computation. Groebner bases are a student's best friend!
 
if by "find" you mean compute, why does "number crunching" seem inappropriate?
 

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