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Normaliz: algorithms for rational cones and affinemonoids
Winfried Bruns
FB Mathematik/InformatikUniversitat Osnabruck
wbruns@uos.de
Osnabruck, January 15, 2013
Winfried Bruns Normaliz: algorithms for rational cones and affine monoids
Overview
1 introductory examples
2 basic definitions and basic theorems
3 tasks of Normaliz
4 the Normaliz algorithm for Hilbert bases5 extensions and modifications6 mastered challenges7 platforms and systems
8 contributors and references
Winfried Bruns Normaliz: algorithms for rational cones and affine monoids
Affine Monoids
Definition
An affine monoid is a finitely generated submonoid of a lattice Zd ,
i.e.,
M � Zd , M C M � M , 0 2 M ,
there exist x1; : : : ; xn 2 M such that
M D fa1x1 C � � � C anxn W ai 2 ZCg:
M is positive if x ; �x 2 M ) x D 0.
Winfried Bruns Normaliz: algorithms for rational cones and affine monoids
A trivial example
0
M D Z2C
(unique) minimal system of generators given by .1; 0/; .0; 1/
Winfried Bruns Normaliz: algorithms for rational cones and affine monoids
A not so trivial example
0
M D fx 2 Z2 W x � 2y ; 3x � yg
(unique) minimal system of generators given by.2; 1/; .1; 1/; .1; 2/; .1; 3/
Winfried Bruns Normaliz: algorithms for rational cones and affine monoids
Cones and lattices
Normaliz computes monoids that arise as intersections of cones andlattices (as the examples above):
Definition
A (rational) cone C is a subset
C D fa1x1 C � � � C anxn W a1; : : : ; an 2 RCg
with a generating system x1; : : : ; xn 2 Zd .
C pointed ” .x ; �x 2 C ) x D 0).
(For us) a lattice is a subgroup of Zd .
We will often assume L D Zd
C pointed and dim C D d .0
Winfried Bruns Normaliz: algorithms for rational cones and affine monoids
Basic theorems: : the support hyperplanes
By the theorem of Minkowski-Weyl finitely generated cones can bedescribed by finitely many inequalities:
Theorem
For C � Rd the following are equivalent:
there are x1; : : : ; xn such that C D fP
aixi W aiRCg;
there are �1; : : : ; �s 2 .Rd/� such thatC D fx 2 R
d W �i.x/ � 0g.
In the case dim C D d (to which we have restricted ourselves)�1; : : : ; �s for minimal s define the support hyperplanes
Hi D fx W �i.x/ D 0g
and the facets Fi D C \ Hi of C.
Winfried Bruns Normaliz: algorithms for rational cones and affine monoids
Basic theorems: the Hilbert basis
x 2 M is irreducible if x D y C z ) x D 0 or y D 0.
Theorem (van der Corput)
Let M be a positive affine monoid.
every element of M is a sum of irreducible elements.
M has only finitely many irreducible elements.
The irreducible elements form the unique minimal system ofgenerators Hilb.M/ of M, the Hilbert basis.
In particular, monoids of type C \ Zd (C pointed, rational) have a
unique minimal finite system of generators, often called Hilb.C/.
Winfried Bruns Normaliz: algorithms for rational cones and affine monoids
Basic theorems: Gordan’s lemma
Theorem (Gordan)
Let C � Rd be the cone generated by x1; : : : ; xn 2 Z
d . Then C \ Zd
is an affine monoid.
Proof.
Let y 2 C \ Zd . Then there exist ai 2 RC such that
y D a1x1 C � � � C anxn:
Write ai D bi C qi mit bi 2 ZC and 0 � qi < 1. Then
y D b1x1 C � � � C bnxn C z; z D q1x1 C � � � C qnxn 2 C \ Zd :
Therefore the monoid C \ Zd is generated by x1; : : : ; xn and the
finite setZ
d \ fq1x1 C � � � C qnxn W 0 � qi < 1g
Winfried Bruns Normaliz: algorithms for rational cones and affine monoids
The tasks of Normaliz: Hilbert basis
Normaliz computes (together with other data)
Hilb.C \ L/
Cones C and lattices L can be specified by
generators x1; : : : ; xn 2 Zd ,
constraints: homogeneous systems of diophantine linearinequalities, equations and congruences,
relations: binomial equations.
Normaliz has two algorithms for Hilbert bases: (1) the originalNormaliz algorithm, and (2) a variant of an algorithm due to Pottier.
We concentrate on generators as input and the algorithm (1).
Winfried Bruns Normaliz: algorithms for rational cones and affine monoids
The tasks of Normaliz: Hilbert series
A grading on M is a surjective Z-linear form deg W gp.M/ ! Z suchthat deg.x/ > 0 for x 2 M , x ¤ 0
The Hilbert (or Ehrhart) function is given by
H.M; k/ D #fx 2 M W deg x D kg
and the Hilbert (Ehrhart) series is
HM.t/ D
1X
kD0
H.M; k/tk :
0
Theorem (Hilbert-Serre, Ehrhart)
HM.t/ is a rational function
H.M; k/ is a quasi-polynomial for k � 0
Winfried Bruns Normaliz: algorithms for rational cones and affine monoids
System of generators and reduction
Most likely every algorithm for the computation of Hilbert basesneeds two phases:
the determination of system of generators E ,
the reduction of E to the Hilbert basis.
We need two auxiliary, interleaved steps:
the computation of the support hyperplanes of the cone,
a triangulation of the cone.
A triangulation is a decompositioninto simplicial cones: C D
S�2˙
C� .
A cone is simplicial if it is generated by linearly independent vectors.
Winfried Bruns Normaliz: algorithms for rational cones and affine monoids
Support hyperplanes: Fourier-Motzkin elimination
This is an incremental algorithm that builds a cone by successiveextending the system of generators and determining the supporthyperplanes in this process.
Start: We may assume that x1; : : : ; xd are linearly independent. Thecomputation of the support hyperplanes is then simply the inversionof the matrix with rows x1; : : : ; xd . (In principle superfluous.)
Extension: we add xdC1; : : : ; xn successively: from the supporthyperplanes of C0 D RCx1 C � � � C RCxn�1 we must compute thesupport hyperplanes of C D C0 C RCxn.
We describe this process geometrically.
Winfried Bruns Normaliz: algorithms for rational cones and affine monoids
We determine the boundary V of the part of C0 that is visible from xn
and its decomposition into subfacets. Together with xn these spanthe new facets of The facets of C0 that are visible from xn
(�i.xn/ < 0), are discarded.
In the cross-section of a 4-dimensional cone:
x1
x2
x3
x4x5
The main problem: find V .
Winfried Bruns Normaliz: algorithms for rational cones and affine monoids
Incremental triangulation
It follows the same inductive scheme, interleaved withFourier-Moatzkin elimination: we obtain a triangulation of C if weextend the triangulation of C0 by the simplicial cones that arespanned by xn and the facets visible from xn.
In the cross-section of a 4-dimensional cone:
Main problem: triangulation may be very large. (Ways out: pyramiddecomposition and partial triangulation.)
Winfried Bruns Normaliz: algorithms for rational cones and affine monoids
Reduction
The reduction is in principle very simple if one knows the supporthyperplanes (or rather the linear forms �1; : : : ; �s).
Theorem
Let E be a system of generators of the positive normal affine monoidM. An element is x 2 M reducible if and only if there exists y 2 E,y ¤ x, such that �i.x � y/ � 0 for i D 1; : : : ; s.
Evidently true, since for x ; y 2 Zd one has x � y 2 C \ Z
d if andonly if �i.x � y/ � 0 for i D 1; : : : ; s.
Main problems:
E is very large and many comparisons are necessary. In thiscase a sophisticated implementation can help to find y quickly.
s is very large.
Winfried Bruns Normaliz: algorithms for rational cones and affine monoids
The steps of the algorithm
The (primal) Normaliz algorithm runs as follows:
1 Input of x1; : : : ; xn, preparatory coordinate transformation.
2 Computation of the support hyperplanes, interleaved with the3 computation of the triangulation ˙ .4 For every cone C� 2 ˙
computation of a system of generators C� \ Zd .�/ and
its reduction to the Hilbert basis HB�
5 reduction ofS
�2˙HB� to Hilb.C \ Z
d/.6 inverse coordinate transformation.
Only step .�/ has not yet been explained. Or has it?
Winfried Bruns Normaliz: algorithms for rational cones and affine monoids
Simplicial cones
Let x1; : : : ; xd be linearly independent and C D RCx1 C : : : RCxd . Inthe proof of Gordan’s lemma we have learnt:
E D fq1x1 C � � � C qdxd W 0 � qi < 1g \ Zd
together with x1; : : : ; xd generate the monoid C \ Zd .
0x1
x2
Winfried Bruns Normaliz: algorithms for rational cones and affine monoids
Easy to see:
Every residue class in Zd=U, U D Zx1 C � � � C Zxd , has exactly one
representative in U.
0x1
x2
Representatives of residue classes can be quickly computed(x1; : : : ; xd ! triangular matrix) and from an arbitrary representativewe obtain the one in E by division with remainder.
Winfried Bruns Normaliz: algorithms for rational cones and affine monoids
Extensions and modifications
Computation of Hilbert series (since 2000),
based on Stanley decompositions: Versions 2.0–2.5 based online shellings, now on a theorem of Koppe–Verdoolaege,
pyramid decomposition “localizing” triangulation (since version2.7),
partial triangulation via “height 1 strategy” for Hilbert basiscomputations(since version 2.5)
Pottiers (dual) algorithm (since version 2.1, builds Csuccessively as an intersection of halfspaces)
parallelization with OpenMP (since version 2.5)
Brand-new: NmzIntegrate computes Lesbesgue integrals ofpolynomials over rational polytopes and generalized Ehrhart seriees.
Winfried Bruns Normaliz: algorithms for rational cones and affine monoids
Challenges mastered
Condorcet effic. linear order 5 � 5 � 3of plurality voting polytope for S6 contigency tables
Reference Schurmann Sturmfels-Welker Ohsugi-Hibi
computation Ehrhart series volume Hilbert basis
embed dimension 24 16 55
dimension 24 16 43
# extreme rays 3928 720 75
# Hilbert basis (25,192) (720) 75
# supp hyperplanes 30 910 306,955
# full triangulation 347,225,775,338 5,745,903,354 (9,248,527,905)
# eval simpl cones 347,225,775,338 102,526,351 448,645
computation time 218:13:55 h 36:45 min 26:52 min
SUN xFire 4450 (parallelization 20 threads), RAM < 2 GB
Winfried Bruns Normaliz: algorithms for rational cones and affine monoids
Platforms and systems
Normaliz (present public version 2.8) runs on
Apple
Linux
MS Windows
Access to Normaliz from
Singular (library by WB and Christof Soger)
Macaulay 2 (package by Gesa Kampf)
CoCoA (John Abbott, Anna Bigatti, Christof Soger)
Sage (optional package by Andrey Novoseltsev)
polymake (Andreas Paffenholz)
Regina (for 3-manifolds by Benjamin Burton)
GUI interface jNormaliz (by V. Almendra and B. Ichim)
Winfried Bruns Normaliz: algorithms for rational cones and affine monoids
Contributors
Robert Koch (1997-2002, implementation in C)
Witold Jarnicki (2003)
Bogdan Ichim (2007–2008, cxompletely new implementation inC++, versions 2.0, 2.1, jNormaliz)
Christof Soger (since 2009)
Gesa Kampf (Macaulay 2 package)
Andreas Paffenholz (polymake interface)
Andrey Novoseltsev (Sage package)
Winfried Bruns Normaliz: algorithms for rational cones and affine monoids
References
W. Bruns and R. Koch, Computing the integral closure of anaffine semigroup. Univ. Iagell. Acta Math. 39 (2001), 59–70.
W. Bruns and J. Gubeladze, Polytopes, rings, and K-theory.Springer 2009.
W. Bruns, R. Hemmecke, B. Ichim, M. Koppe, and C. Soger,Challenging computations of Hilbert bases of cones associatedwith algebraic statistics. Exp. Math. 20 (2011), 1–9.
W. Bruns and B. Ichim, Normaliz: algorithms for affine monoidsand rational cones. J. Algebra 324 (2010), 1098–1113.
W. Bruns and G. Kampf, A Macaulay 2 interface for Normaliz.Preprint. J. Softw. Algebr. Geom. 2 (2010), 15–19.
W. Bruns, B. Ichim and C. Soger, The power of pyramiddecomposition in Normaliz. Submitted.
W. Bruns and C. Soger, computation of generalized Ehrhartseries in Normaliz. Submitted.
Winfried Bruns Normaliz: algorithms for rational cones and affine monoids
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