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characteristic classes of coincident root loci

# 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

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]
``````