### D.14.1 arr_lib

Library:
  arr.lib
Purpose:
  a library of algorithms for arrangements of hyperplanes

Authors:
  Randolf Scholz (rscholz@rhrk.uni-kl.de),
  Patrick Serwene (serwene@mathematik.uni-kl.de),
  Lukas Kuehne (lf.kuehne@gmail.com)

Overloads:
  // OPERATORS
  = arrAdd assignment
  + arrAdd union of two arrs
  [ arrGet access to a single/multiple hyperplane(s) - arrMinus deletes given hyperplanes from the arr <= arrLEQ comparison
  >= arrGEQ comparison
  == arrEQ comparison
  != arrNEQ comparison
  < arrLNEQ comparison
  > arrGNEQ comparison

  // TYPECASTING
  matrix arr2mat coeff matrix
  poly arr2poly defining polynomial

  // OTHER
  variables arrVariables ideal generated by the variables the arr depends on nvars arrNvars number of variables the arr depends on delete arrDelete deletes hyperplanes by indices print arrPrint prints the arr on the screen

  // IDEAL INHERITED FUNCTIONS
  homog arrHomog checks if arrangement is homogeneous simplify arrSimplify simplifies arrangement size arrSize number of planes
  subst arrSubst substitute variables

  // MULTI-ARRANGEMENTS
  = multarrAdd assignment of multarr
  + multarrAdd union of multarr
  poly multarr2poly defining polynomial
  size multarrSize number of hyperplanes with mult. print multarrPrint displays multiarr
  delete multarrDelete deletes hyperplane

Procedures:

> D.14.1.1 arrSet |  | replaces the k-th Hyperplane with poly p
> D.14.1.2 type2arr |  | converts general input to 'arr' using arrAdd.
> D.14.1.3 mat2arr |  | affine arrangement from coeff matrix
> D.14.1.4 mat2carr |  | central arrangement from coeff matrix
> D.14.1.5 arrPrintMatrix |  | readable output as a coeff matrix
> D.14.1.6 varMat |  | matrix of the corresponding ring_variables
> D.14.1.7 varNum |  | number of given variable (enh. version of varNum in dmod.lib)
> D.14.1.8 arrSwapVar |  | swaps two variables in the arrangement
> D.14.1.9 arrLastVar |  | ring_variable of largest index used in arrangement
> D.14.1.10 arrCenter |  | computes center of an arrangement
> D.14.1.11 arrCentral |  | checks if arrangement is central
> D.14.1.12 arrCentered |  | checks if arrangement is centered
> D.14.1.13 arrCentralize |  | makes centered arrangement central
> D.14.1.14 arrCoordChange |  | performs coordinate change
> D.14.1.15 arrCoordNormalize |  | performs projection onto coordinate hyperplane
> D.14.1.16 arrCone |  | coned arrangement
> D.14.1.17 arrDecone |  | deconed arrangement
> D.14.1.18 arrLocalize |  | localization of an arrangement onto a flat
> D.14.1.19 arrRestrict |  | restricted arrangement onto a flat
> D.14.1.20 arrIsEssential |  | checks if arrangement is essential
> D.14.1.21 arrEssentialize |  | essentialized arragnement
> D.14.1.22 arrBoolean |  | boolean arrangement
> D.14.1.23 arrBraid |  | braid arrangement
> D.14.1.24 arrTypeB |  | type B arrangement
> D.14.1.25 arrTypeD |  | type D arrangement
> D.14.1.26 arrRandom |  | random (affine) arrangement
> D.14.1.27 arrRandomCentral |  | random central arrangement
> D.14.1.28 arrEdelmanReiner |  | Edelman-Reiner arrangement
> D.14.1.29 arrOrlikSolomon |  | Orlik-Solomon algebra of the arrangement
> D.14.1.30 arrDer |  | module of derivation
> D.14.1.31 arrIsFree |  | checks if arrangement is free
> D.14.1.32 arrExponents |  | exponents of a (free) arrangement
> D.14.1.33 arr2multarr |  | converts normal arrangement to multiarrangement
> D.14.1.34 multarr2arr |  | converts multiarrangement to normal arrangement
> D.14.1.35 multarrRestrict |  | restriction of A (as arr) to a flat with multiplicities
> D.14.1.36 multarrMultRestrict |  | restriction of A (as multarr) to a hyperplane with multiplicities
> D.14.1.37 arrFlats |  | intersection lattice
> D.14.1.38 arrLattice |  | computes the intersection lattice / poset
> D.14.1.39 moebius |  | computes moebius values
> D.14.1.40 arrCharPoly |  | characteristic polynomial
> D.14.1.41 arrPoincare |  | poincare polynomial of the arrangement
> D.14.1.42 arrChambers |  | number of chambers of the arrangement
> D.14.1.43 arrBoundedChambers |  | number of bounded chambers of the arrangement
> D.14.1.44 printMoebius |  | displays the moebius values of all the flats in the poset

User manual for Singular version 4.4.1, 2025, generated by texi2html.