Random points on an algebraic manifold

Abstract

Consider the set of solutions to a system of polynomial equations in many variables. An algebraic manifold is an open submanifold of such a set. We introduce a new method for computing integrals and sampling from distributions on algebraic manifolds. This method is based on intersecting with random linear spaces. It produces i.i.d. samples, works in the presence of multiple connected components, and is simple to implement. We present applications to computational statistical physics and topological data analysis.

Voronoi cells of varieties

“…For instance, for the quadratic Veronese surface in P 5 we have e = 2 and hence δ X (y) = 16. This is smaller than the number 18 found in Example 3.4, since back then we were dealing with the cone over the Veronese surface in R 6 , and not with the Veronese surface in R 5 ⊂ P 5 .…”

“…The factor 1 − v is the Euler number of P 1 \{v+1 points}. We will use (5) to derive the degree formulas from Section 5, and refer to [12, for Euler numbers of smooth curves, surfaces and hypersurfaces.…”

“…, E 4d−4 over the remaining base points. The strict transforms of the curves X (t:u) on X are then the fibers of a morphism q : X → P 1 for which we apply (5).…”

“…This quantity is of interest in topological data analysis, where it determines the density of sample points needed to compute the persistent homology of X. We refer to [3,5,10] for recent progress on sampling at the interface of topological data analysis with metric algebraic geometry. The distance from a point y on X to the medial axis could be considered the local reach of X. Equivalently, this is the distance from y to the boundary of its Voronoi cell Vor X (y).…”