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WhitneyStratifications :: WhitneyStratifications

WhitneyStratifications -- Computes Whitney Statifications of real and complex varieties and of algebraic maps between them.

Description

This package computes Whitney stratifications of real and complex algebraic varieties using the algorithms described in [1, 2]. For varieties considered over the complex numbers the output is indexed by the strata dimension. When wishing to treat the variety over the reals, the same output may be used, but the dimensions of the strata may differ (and some strata may be empty), see [2] for more details. This post processing in the real case is currently left to the user.

A method is also provided to stratify polynomial maps $f:X\to Y$ between algebraic varieties, the output is a Whitney stratification of both $X$ and $Y$, such that for each (open, connected) strata $M$ of $X$ there is an (open, connected) strata $N$ of $Y$ such that $f(M) \subset N$ and such that the restriction of $f$ to $M$ is a submersion.

Computing the Conormal variety of a variety is an important step in these algorithms, so a method for this is also provided.

References:

[1] Martin Helmer and Vidit Nanda. "Conormal Spaces and Whitney Stratifications", Foundations of Computational Mathematics, DOI: 10.1007/s10208-022-09574-8.

[2] Martin Helmer and Vidit Nanda "Effective Whitney stratification of real algebraic varieties", arXiv:2307.05427v2, 2023.

Author

Version

This documentation describes version 2.03 of WhitneyStratifications.

Source code

The source code from which this documentation is derived is in the file WhitneyStratifications.m2.

Exports

  • Functions and commands
    • conormal -- Computes the conormal variety
    • conormalRing (missing documentation)
    • mapStratify -- Computes a Whitney stratification of a polynomial map $f:X\to Y$ between real or complex varieties.
    • nonProperSet (missing documentation)
    • whitneyStratify -- Computes a Whitney stratification of the real and complex varieties.
  • Methods
    • "conormal(Ideal)" -- see conormal -- Computes the conormal variety
    • "mapStratify(List,Ideal,Ideal)" -- see mapStratify -- Computes a Whitney stratification of a polynomial map $f:X\to Y$ between real or complex varieties.
    • "whitneyStratify(Ideal)" -- see whitneyStratify -- Computes a Whitney stratification of the real and complex varieties.
  • Symbols
    • isProper (missing documentation)
    • StratsToFind (missing documentation)

For the programmer

The object WhitneyStratifications is a package.