Skip to content
December 18, 2013 / porton

A negative result on a conjecture

Due my research about singularities the problem formulated in this blog post was solved negatively with help of Alex Ravsky who has found a counter-example.

The conjecture was: \mathrm{GR}(\Delta \times^{\mathsf{FCD}} \Delta) is closed under finite intersections.

The counter-example follows: f=\{(x,y)\in\mathbb R^2:|x|\le |y| \vee y=0\}, g=\{(x,y)\in\mathbb R^2:|x|\ge |y| \vee x=0\}.

It is easy to show that f,g\in \mathrm{GR}(\Delta \times^{\mathsf{FCD}} \Delta) but f\cap g\notin \mathrm{GR}(\Delta \times^{\mathsf{FCD}} \Delta).

This result is discouraging, because (as it seems) it probably follows that using plain funcoids approach to singularity theory fails and we need to invent something more sophisticated.

Also note that f,g\in \mathrm{GR}(\Delta \times^{\mathsf{FCD}} \Delta) is a rather counter-intuitive result (draw a graph to see it), despite of the fact that it can be easily proved.


Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

This site uses Akismet to reduce spam. Learn how your comment data is processed.

%d bloggers like this: