This package can be used to make experiments, trying many ideals, perhaps over small fields. For example...what would you expect the regularities of "typical" monomial ideals with 10 generators of degree 3 in 6 variables to be? Try a bunch of examples -- it’s fast. Here we do only 500 -- this takes about a second on a fast machine -- but with a little patience, thousands can be done conveniently.

i1 : setRandomSeed(currentTime()) o1 = 1575146618 |

i2 : kk=ZZ/101; |

i3 : S=kk[vars(0..5)]; |

i4 : time tally for n from 1 to 500 list regularity randomMonomialIdeal(10:3,S) -- used 0.800975 seconds o4 = Tally{3 => 1 } 4 => 39 5 => 192 6 => 200 7 => 57 8 => 9 9 => 2 o4 : Tally |

How does this compare with the case of binomial ideals? or pure binomial ideals? We invite the reader to experiment, replacing "randomMonomialIdeal" above with "randomBinomialIdeal" or "randomPureBinomialIdeal", or taking larger numbers of examples. Click the link "Finding Extreme Examples" below to see some other, more elaborate ways to search.

- Finding Extreme Examples -- Ways to use random ideals to search for (counter)-examples
- randomIdeal -- randomIdeal made from a given set of monomials
- randomMonomialIdeal -- random monomial ideal with given degree generators
- randomSquareFreeMonomialIdeal -- random square-free monomial ideal with given degree generators
- randomSquareFreeStep -- A step in a random walk with uniform distribution over all monomial ideals
- randomBinomialIdeal -- randomBinomialIdeal with binomials of given degrees
- randomPureBinomialIdeal -- randomPureBinomialIdeal with binomials of given degrees
- randomSparseIdeal -- randomSparseIdeal made from a given set of monomials
- randomElementsFromIdeal -- Chooses random elements of given degrees in a given ideal.
- randomMonomial -- Choose a random monomial of given degree in a given ring
- randomShellableIdeal -- Produces a ideal from a random shellable simplicial complex
- randomShellableIdealChain -- Produces a chain of ideals from a random shelling
- randomShelling -- produces a random chain of shellable complexes

- Katie Ansaldi <kansaldi@gmail.com>
- Jay Yang <jkelleyy@gmail.com>

- Functions and commands
- idealChainFromShelling -- Produces chains of ideals from a shelling.
- idealFromShelling -- Produces an ideal from a shelling
- isShelling -- determines whether a list represents a shelling of a simplicial complex.
- randomAddition -- Adds a random facet to a shellable complex
- randomBinomialEdgeIdeal -- Creates a binomial edge ideal from a random graph with n vertices and t edges.
- randomBinomialIdeal -- randomBinomialIdeal with binomials of given degrees
- randomEdgeIdeal -- Creates an edge ideal from a random graph with n vertices and t edges.
- randomElementsFromIdeal -- Chooses random elements of given degrees in a given ideal.
- randomIdeal -- randomIdeal made from a given set of monomials
- randomMonomial -- Choose a random monomial of given degree in a given ring
- randomMonomialIdeal -- random monomial ideal with given degree generators
- randomPureBinomialIdeal -- randomPureBinomialIdeal with binomials of given degrees
- randomShellableIdeal -- Produces a ideal from a random shellable simplicial complex
- randomShellableIdealChain -- Produces a chain of ideals from a random shelling
- randomShelling -- produces a random chain of shellable complexes
- randomSparseIdeal -- randomSparseIdeal made from a given set of monomials
- randomSquareFreeMonomialIdeal -- random square-free monomial ideal with given degree generators
- randomSquareFreeStep -- A step in a random walk with uniform distribution over all monomial ideals
- randomToricEdgeIdeal -- Creates a toric edge ideal from a random graph with n vertices and t edges.
- regSeq -- regular sequence of powers of the variables, in given degrees
- squareFree -- ideal of all square-free monomials of given degree

- Symbols
- AlexanderProbability -- option to randomSquareFreeStep