Project: Numerical Decomposition

• Current people involved: Tim Duff, Anton Leykin, Jose Rodriguez
• Goal: implement numerical irreducible decomposition for multiprojective varieties and related tools
• Current status: under development

Project Description

Give a variety embedded in a product of projective spaces that is given by a system of multihomogeneous polynomial equations, we would like to describe it using (collections) of witness sets. (A witness set is a cornerstone concept of numerical algebraic geometry.)

The current plan is to

• implement most of the tools described in [1],
• revamp the current implementation (in NumericalAlgebraicGeometry package) of decomposition in the case of ambient affine/projective space (a single factor case) using u-generation in [2],
• work on multiprojective u-generation in [2].

Examples

The old implementation of the numerical decomposition of an affine variety:

i1 : needsPackage "NumericalAlgebraicGeometry";
...
...
...
i5 : R = CC[x,y,z];
i6 : sph = x^2+y^2+z^2-1;
i7 : I = ideal {x*sph*(y-x^2), sph*(z-x^3)};  
o7 : Ideal of R
i8 : numericalIrreducibleDecomposition I   
o8 = a numerical variety with components in      
     dim 1:  (dim=1,deg=1) (dim=1,deg=3)      
     dim 2:  (dim=2,deg=2)  
o8 : NumericalVariety

References (that describe the math and/or algorithms)

[1] A numerical toolkit for multiprojective varieties

Authors: Jonathan D. Hauenstein, Anton Leykin, Jose Israel Rodriguez, Frank Sottile

[2] u-generation: solving systems of polynomials equation-by-equation

Authors: Timothy Duff, Anton Leykin, Jose Israel Rodriguez