RandomComplexes -- support for creating random complexes over the integers

Description

We implement two methods to create a random ChainComplex over the integers. The first method (randomChainComplex) builds the complex from products of randomly chosen matrices of desired rank. The limitation of this method to produce large complexes over the integers with moderate Height is the use of the LLL algorithm to improve the presentation of syzygy matrices.

The second method (randomSimplicialComplex) uses Stanley-Reisner rings from randomly chosen monomial ideals.

Caveat

Some functionality here should be moved elsewhere, e.g. disturb, histogram, maximalEntry, and normalize.

Authors

Version

This documentation describes version 0.2 of RandomComplexes.

Citation

If you have used this package in your research, please cite it as follows:

@misc{RandomComplexesSource,
  title = {{RandomComplexes: random complexes over fields or the integers. Version~0.2}},
  author = {Frank-Olaf Schreyer and Michael E. Stillman},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/master/M2/Macaulay2/packages}}
}

Exports

Functions and commands
- disturb -- disturb the matrices of a chain complex over RR
- histogram -- histogram of a list of real numbers
- maximalEntry -- maximal absolute value of the entries of the matrix or matrices
- normalize -- normalize a ChainComplex over QQ or RR
- randomChainComplex -- random chain complex over the integers with prescribed ranks of the homology group and ranks of the matrices
- randomSimplicialComplex -- the chainComplex over ZZ of a random Stanley-Reisner simplicial complex
- testTimeForLLLonSyzygies -- test timing for LLL on syzygies
Methods
- disturb(ChainComplex,RR) -- see disturb -- disturb the matrices of a chain complex over RR
- histogram(List,ZZ) -- see histogram -- histogram of a list of real numbers
- maximalEntry(ChainComplex) -- see maximalEntry -- maximal absolute value of the entries of the matrix or matrices
- maximalEntry(Matrix) -- see maximalEntry -- maximal absolute value of the entries of the matrix or matrices
- normalize(ChainComplex) -- see normalize -- normalize a ChainComplex over QQ or RR
- randomChainComplex(List,List) -- see randomChainComplex -- random chain complex over the integers with prescribed ranks of the homology group and ranks of the matrices
- randomSimplicialComplex(ZZ,ZZ) -- see randomSimplicialComplex -- the chainComplex over ZZ of a random Stanley-Reisner simplicial complex
- testTimeForLLLonSyzygies(ZZ,ZZ) -- see testTimeForLLLonSyzygies -- test timing for LLL on syzygies
Symbols
- Continuous -- see Discrete -- Value for the Strategy in disturb
- Discrete -- Value for the Strategy in disturb
- WithLLL -- Option for randomComplex
- ZeroMean -- Option for randomComplex

For the programmer

The object RandomComplexes is a package, defined in RandomComplexes.m2.

The source of this document is in RandomComplexes.m2:248:0.