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GlicciPointsInP3 :: GlicciPointsInP3

GlicciPointsInP3 -- Experiments establishing direct Gorenstein linkages for finite subsets of P^3

Description

We determine for which pairs of integers (d,e) there exist a direct Gorenstein link from the generic set of d points in P^3 to a generic set of e points in P^3. Our method of proving existence is computational: we choose examples of Gorenstein linkages using a probabilstic method, and show by a tangent space computation that the incidence correspondence are bidominant. Our paper Twenty Points in P^3 shows that no further bidominant correspondences can occur, and explains how the routines in this package establish the existence.

Splitting and Factoring

Degrees and h-Vectors

  • hVector -- compute the h-vector
  • BEdegrees -- compute the degrees of the Buchsbaum-Eisenbud skew matrix

Tangent Space Computations

  • tangentDimension -- compute the dimension of tangentspaces
  • dimFormula -- return dimension of HC_hV
  • checkDimensionsHCG -- check that the dimension formula gives the dimension of the tangentspace at HC_G for a random choice of G

Building the Bidominance Graph

  • listhVectors -- list all admissible h-vectors of Gorenstein points up to b
  • createCandidates -- create numerical candidates for bi-dominant Gorenstein linkage correspondences
  • checkCandidates -- check which numerical candidates lead to a bi-dominant Gorenstein linkage correspondence
  • getGraph -- compute the graph of the bi-dominant Gorenstein linkage correspondences.
  • graphvizFile -- builds an input file for graphviz

Univariate Polynomial Statistics

  • isEven -- Is L the class of an even permutation?
  • sizeConjClass -- number of permutations with given cycle decomposition
  • probOfFactor -- probability that a polynomial of degree n is square free AND has a factor of degree k over the same finite ground field
  • primeDivisors -- compute List of prime factors of an integer
  • numOfIrreducible -- the number of irreducible polynomials
  • numOfSquareFree -- the number of irreducible polynomials
  • polynomialsWithFactor -- number of monic square-free polynomials of degree n with factor of degree degree k over F_q

Verification

Authors

Version

This documentation describes version 0.3 of GlicciPointsInP3.

Source code

The source code from which this documentation is derived is in the file GlicciPointsInP3.m2.

Exports

  • Functions and commands
    • BEdegrees -- compute the degrees of the Buchsbaum-Eisenbud skew matrix
    • checkCandidates -- check which numerical candidates lead to a bi-dominant Gorenstein linkage correspondence
    • checkDimensionsHCG -- check that the dimension formula gives the dimension of the tangentspace at HC_G for a random choice of G
    • createCandidates -- create numerical candidates for bi-dominant Gorenstein linkage correspondences
    • dimFormula -- return dimension of HC_hV
    • findSplittingExample -- find an example of a Gorenstein ideal of points with a subset of degree n
    • getGraph -- compute the graph of the bi-dominant Gorenstein linkage correspondences.
    • graphvizFile -- builds an input file for graphviz
    • hasFactor -- check for a factor of desired degree
    • hVector -- compute the h-vector
    • isEven -- Is L the class of an even permutation?
    • listhVectors -- list all admissible h-vectors of Gorenstein points up to b
    • numOfIrreducible -- the number of irreducible polynomials
    • numOfSquareFree -- the number of irreducible polynomials
    • pickGorensteinIdeal -- select a codimension 3 Gorenstein ideal
    • polynomialsWithFactor -- number of monic square-free polynomials of degree n with factor of degree degree k over F_q
    • primeDivisors -- compute List of prime factors of an integer
    • probOfFactor -- probability that a polynomial of degree n is square free AND has a factor of degree k over the same finite ground field
    • sizeConjClass -- number of permutations with given cycle decomposition
    • tangentDimension -- compute the dimension of tangentspaces
    • testSplitting -- test a random Gorenstein set of points in P^3 for a factor
    • verifyAllAssertionsOfThePaper -- print commands to verfy the assertions of the paper
    • verifyGraphInSmallCharacteristic -- print commands to verfy the assertions of the paper