Singular

D.14.1 arr_lib

Library:

arr.lib

Purpose:

a library of algorithms for arrangements of hyperplanes

Authors:

Randolf Scholz ([email protected]),
Patrick Serwene ([email protected]),
Lukas Kuehne ([email protected])

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