A Package for Linear Diophantine Systems, Polyhedra and Lattices
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Neues uber¨ Systeme Normaliz: A package for linear diophantine systems, polyhedra and lattices W. Bruns (Universitat¨ Osnabruck,¨ Institut fur¨ Mathematik, Germany) [email protected] d The very first version of Normaliz [5] was meant where C is a pointed cone in R and L is a sublattice of d to compute normalizations of affine monoids (or semi- Z . In our case d = 9, C is the positive orthant, and L is groups), hence the name. Over the years it has been ex- the lattice of integer solutions to the system (6). There tended to a powerful package for discrete convex geom- are other ways to describe M, but this choice of C and etry. We explain its main computation goals by a simple L is exactly Normaliz’ approach to the problem. Two example, sketch the mathematical background and dis- natural questions suggest themselves: cuss the basic steps in the Normaliz primal algorithm. Some remarks on the technical aspects and the history (Generation) Is the monoid of magic squares conclude this overview. finitely generated, and if so, what is a system of generators? The main computation goals (Enumeration) Given M, how many magic squares of magic constant M are there? Suppose we are interested in the following type of 3 × 3 matrices The answers to Generation and Enumeration are the main computation goals of Normaliz. x1 x2 x3 x4 x5 x6 x7 x8 x9 Some background and the problem is to find nonnegative integer values The introductory example is a rather special case of the for x ; : : : ; x such that the 3 numbers in all rows, all 1 9 objects for which Normaliz solves Generation and Enu- columns, and both diagonals sum to the same constant meration: since version 3.0 it can be applied to arbitrary M. Sometimes such matrices are called magic squares intersections of rational polyhedra and affine lattices. In and M is the magic constant. This leads to a system of other words, Normaliz is a solver for systems of affine- 7 diophantine linear equations: linear diophantine inequalities, equations and congru- x + x + x = x + x + x ; ences. 1 2 3 4 5 6 Nevertheless, for the sake of simplicity, in this ::: = ::: (6) overview we stick to the special case M = C \ L, and x1 + x2 + x3 = x3 + x5 + x7: specialize it even further, following Normaliz’ course of computation. In our example we can first introduce co- Our goal is to understand the set of solutions in nonneg- ordinates in the solution space of the system (6), and ative integers. then intersect the positive orthant with it. After this Since the equations are linear homogeneous, and the reduction step, we can assume that we want to com- positive orthant is closed under addition, the sum of two pute the integer points in a d-dimensional pointed cone d solutions is again a solution (the magic constants add C ⊂ R . This is the core case to which Normaliz re- up). It follows that the set M of all solutions is a monoid duces the given input data by preliminary transforma- (or semigroup). In a systematic framework we can de- tions. Let us first solve Generation for cones: scribe it as follows: d Theorem 1 (Minkowski-Weyl) Let C ⊂ R . Then the M = C \ L following are equivalent: 12 d 1. There exist (integer) vectors x1; : : : ; xn 2 R Compute the Hilbert basis of a rational cone C. such that The Hilbert basis is the set of irreducible elements in d C = fx 2 R : x = α1x1 + ··· + αnxn; M, i.e., elements x that can not be written in the form α : : : : ; α ≥ 0g: x = y + z with y; z 2 M, y; z 6= 0. This proves unique- 1 n ness, and is important for the reduction of a generating 2. There exist linear forms λ1; : : : ; λs (with integer set to the Hilbert basis. d coefficients) on R such that Let us turn to Enumeration. For it we need a grad- ing, a -linear form deg : d ! . For the next theo- C = fx 2 d : λ (x) ≥ 0; i = 1; : : : ; sg: Z Z Z R i rem we restrict the generality even further. If the equivalent conditions of the theorem are sat- Theorem 3 (Hilbert; Ehrhart) Let d ≥ 1. Suppose isfied, C is called a (rational) cone. (Sometimes the that the extreme rays of the d-dimensional cone C have attribute polyhedral is added.) The cone C is pointed degree 1 and set M = C \ d. Then the lattice point if it does not contain a linear subspace of positive di- Z enumerator mension. In this case the elements of a minimal set x1; : : : ; xn of generators are unique up to permutation H(M; k) = #fx 2 M : deg x = 1g and multiplication by positive scalars. We call them ex- treme rays. The scalar is 1 in the integer case if we re- is given by a polynomial qM with rational coefficients quire that the coefficients are coprime. for all k ≥ 0. Equivalently, the generating function P1 k The dimension of a cone is the dimension of the vec- k=0 H(M; k)t defines a rational function of type tor subspace it generates. If dim C = d, then the lin- ear forms λ are unique up to permutation and positive 1 + h t + ··· + h tu i H (t) = 1 u ; scalar multiples if the system is minimal, and again the M (1 − t)d scalar must be 1 if we require that their coordinates are h1; : : : ; hu 2 Z; u < d: coprime integers. The linear forms λ1; : : : λs then define the (relevant) support hyperplanes H = fx : λ (x) = i i The polynomial qM is called the Hilbert polynomial 0g, and Theorem 1(2) represents C as an intersection of and HM (t) the Hilbert series of M – for very good rea- the (positive) halfspaces H+ = fx : λ (x) ≥ 0g. λi i son since the theorem can be derived from the theory The conversion from generators to support hyper- of graded algebras, a key observation of Stanley. In the planes is usually called convex hull computation and the combinatorial context it was independently proved by converse is called vertex enumeration. Both directions Ehrhart, and therefore one speaks of the Ehrhart poly- are completely equivalent since they amount to the du- nomial and Ehrhart series as well. The hi are nonnega- alization of a cone. tive; this follows from Hochster’s theorem by which the That Generation makes sense also for lattice points monoid algebra K[M] is Cohen-Macaulay for any field is guaranteed by K, but it can also be shown combinatorially. d Theorem 2 (Gordan’s lemma) Let C ⊂ R be a ra- Without the hypothesis on the degree of the extreme d tional pointed cone. Then the monid M = C \ Z is rays, the theorem must be reformulated: the polynomial finitely generated. More precisely, it has a unique mini- is only a quasipolynomial in general, i.e., a “polyno- mal system E of generators. mial” with coefficients that are periodic in k, and the denominator takes a more complicated form. A set fz1; : : : ; zmg ⊂ M generates the monoid The leading coefficient of the Hilbert/Ehrhart poly- M if every element of M is a linear combination of nomial qM has the form e(M)=(d − 1)! with a positive z1; : : : ; zm with nonnegative integer coefficients. The integer e(M) that is called the multiplicity of M. It is the unique minimal generating set Hilb(C) (or Hilb(M)) lattice normalized volume of the polytope spanned by in Theorem 2 is called the Hilbert basis the extreme rays. Moreover, e(M) = 1 + h1 + ··· + hu. Clearly Compute the Hilbert series of M is the right formulation of Enumeration now. The magic squares continued 0 The input to Normaliz for the computation of the magic Figure 1: A Hilbert basis squares is encoded as follows: amb_space 9 of C (or M). We do not see a good historical reason equations 7 for this nomenclature – the term “Gordan basis” would 1 1 1 -1 -1 -1 0 0 0 be much more appropriate. In the restricted setting that ... we have reached, we can now reformulate Generation 1 1 0 0 -1 0 -1 0 0 grading as follows: 1 1 1 0 0 0 0 0 0 13 The first equation reads in reasonable time, but the Hilbert series for 6 × 6 is out of reach. x1 + x2 + x3 − x4 − x5 − x6 = 0; as desired, and the other equations are encoded analo- The primal algorithm gously. The magic constant is the grading. If only equa- tions are given, but no inequalities, Normaliz assumes For the computation of Hilbert bases Normaliz provides that the nonnegative solutions should be computed. We two algorithms at the user’s disposal. Here we restrict look at some data in the output file: ourselves to the primal algorithm that also computes the Hilbert series if this is desired. For the dual algorithm 5 Hilbert basis elements we refer the reader to Bruns and Ichim [3]. 5 Hilbert basis elements of degree 1 4 extreme rays The primal algorithm is based on triangulations. We 4 support hyperplanes assume that the cone C is pointed and defined by a gen- embedding dimension = 9 erating set. (If it is defined by inequalities, Normaliz rank = 3 first computes the extreme rays.) After some prelimi- ... grading: nary transformations we can further assume that C has d 1 1 1 0 0 0 0 0 0 dimension d and that Z is the lattice to be used. Then with denominator = 3 the primal algorithm proceeds in the following steps: The input grading is the magic constant.