Many problems in science can be described by polynomial equations. The solution set of the corresponding
polynomial system is referred to as an algebraic geometrical model for the problem.
When the solution set consists of isolated points the model is easy to describe and even to visualise.
For higher dimensional solution spaces, deeper and more
sophisticated geometrical and numerical techniques are required. Some ideas for algebraic sampling
and on how to estimate its density will be presented. The key challenge is to estimate
the right density to recover the topological signature of the model.
Bottlenecks are pairs of points on a variety joined by a line which is normal to the variety
at both points. These points play a special role in determining the appropriate density of a point-sample
of the variety. Under suitable genericity assumptions the number of bottleneck of an affine variety
is finite and we call it the bottleneck degree. We will show that it is determined by invariants of the variety.
The talk is based on joint work with D. Eklund and M. Weinstein.