Goals:
(W) implement (more orderly) data types representing (equidimensional) components of varieties
(M) support multiprojective geometry
(D) make robust the nonreduced case
(W) is about various witness sets, here are the proposed types:
* WSet should layout a framework of methods ("primitives") that are used by higher level algorithms
* PWSet, a projective witness set. Same for a component of a projective variety.
* PWSet should link to one or several WSets, witness sets in particular affine charts
* A user option "Projectivize" should force an answer to an affine problem to be accompanied by an answer to its projectivization.
* ProxyWSet should represent an image of a rational map.
There should be either a field or a method called "ambient" for all of the above.
A NumericalVariety then is a collection of witness sets that have the same ambient.
(M) MPWSet should represent a component of a multiprojective variety.
* Should it be a "collection of witness sets"?... or "collection" should be its own type?
* is MWSet, the specialization to a particular set or affine charts, necessary?
(D) for deflation.
* deflation of an isolated point.
* deflation of a positive-dimensional variety: the output should be expressed in terms of ProxyWSet.
* should we attach a "multiplicity structure"? (e.g., expressed in terms of an inverse system, via a DualSpace, etc.)
Examples:
* A kind of ProxyWSet is already in use by NumericalImplicitization.
* A biprojective ProxyWSet comes out of positive-dimensional deflation.
* A biprojective ProxyWSet appears (in the background) of the MonodromySolver approach to finding a generic (0-dim) fiber of a covering.
Questions:
* Given two NumericalVarieties what is a good way to compute/describe their intersection?
* In multiprojective geometry, MPWSet for an irreducible component is supported on a polytope (in the lattice of "multidimensions").
In the context of MWSet (i.e., a problem that comes as a multiprojective problem specialized to a certain collection of affine charts),
can we say anything nice about the multidegree? (apart from the observation that it is bounded by the projective multidegree...
that in turn may be bounded by some generalization of the log-concavity statement).
* Should we store DeflationSystem and DeflationRandomMatrix(matrices) with a system? or a point?