Characteristic classes of coincident root loci
==============================================
Coincident root loci (or discriminant strata) are subsets
of the space of homogeneous polynomials in two variables defined by
root multiplicities: A nonzero degree _n_ polynomial has _n_ roots in the
complex projective line P^1, but some of these can coincide, which gives us a
partition of _n_. Hence for each partition _lambda_ we get a set of polynomials
(those with root multiplicities given by _lambda_),
which together stratify the space of these polynomials, which (modulo multiplying by scalars)
is P^n. These are quasi-projective varieties, invariant under the action of GL(2);
their closures are highly singular projective varieties, making them a good
example for studying invariants of singular varieties.
This package contains a number of different algorithms to compute invariants
and characteristic classes of these varieties:
- degree
- Euler characteristic
- the fundamental class in equivariant cohomology
- Chern-Schwartz-MacPherson (CSM) class, Segre-SM class
- equivariant CSM class
- Hirzebruch Chi-y genus
- Todd class, motivic Hirzebruch class
- motivic Chern class
- equivariant motivic Chern class
Some of the algorithms are implemented in Mathematica
instead of (or in addition to) Haskell.
Another (better organized) Mathematica implementation is available at
<https://github.com/bkomuves/mathematica-packages>.
Example usage
=============
For example if you want to know what is the equivariant CSM class of the
(open) loci corresponding to the partition [2,2,1,1], you can use the following
piece of code:
{-# LANGUAGE TypeApplications #-}
import Math.Combinat.Partitions
import Math.RootLoci.Algebra.SymmPoly ( AB )
import Math.Algebra.Polynomial.Pretty ( pretty )
import Math.RootLoci.CSM.Equivariant.Umbral
csm ps = umbralOpenCSM @AB (mkPartition ps)
main = do
putStrLn $ pretty $ csm [2,2,1,1]