characteristic classes of coincident root loci

## root

# 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]
```