Singular

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)

// OPERATORS
+ 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
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