The computation of generalized Ehrhart series and integrals in Normaliz

Transcription

1 The computation of generalized Ehrhart series and integrals in Normaliz FB Mathematik/Informatik Universität Osnabrück Berlin, November 2013

2 Normaliz and NmzIntegrate The computer program Normaliz has been developed in Osnabrück since Current team members: Bogdan Ichim (Bucharest), Christof Söger (OS), WB Now supported by the DFG SPP Experimental methods in algebra, geometry and number theory Normaliz computes two types of data: 1 Hilbert bases of rational cones (normalizations of affine monoid domains) 2 Ehrhart series of rational polytopes (Hilbert series of normal monoid domains) Normaliz has an offspring NmzIntegrate (by WB and C. Söger) for the computation of generalized (or weighted) Ehrhart series. NmzIntegrate is based on CoCoALib.

3 Ehrhart series P R d rational polytope. E.P; k/ D #.kp \ Z d / D # P \ 1 k Zd is called the Ehrhart function of P. The generating function E P.t/ D 1X E.P; k/t k kd0 is the Ehrhart series. Example: Q R 2 unit square. Then Q 2Q E.Q; k/ D.k C 1/ 2 E Q.t/ D 1 C t.1 t/ 3

4 Ehrhart series as Hilbert series Data: C pointed polyhedral rational cone in R d, monoid M D C \ Z d, grading deg W gp.m/! Z surjective, deg x > 0 for x 2 M, x 0 deg C P P Set E M.t/ D P x2m t deg x. Then E M.t/ D E P.t/ D H KŒM.t/ P D fy 2 C W deg y D 1g K a field, H Hilbert series (deg extended to RM)

5 Generalized Ehrhart series M Z d a graded monoid as above, f a polynomial in d variables (with rational coefficients). Then E M;f.t/ D X x2m f.x/t deg x is the f -generalized (or f -weighted) Ehrhart series of M. For rational polytopes E.P; f; k/ D X x2kp\z d f.x/; E P;f.t/ D 1X E.P; f; k/t k : kd0 Typical application: W R d! R e is a projection, P R d, Q D.P/, f.x/ D # 1.x/. Then E P.t/ D E Q;f.t/: If dim Q dim P, this is usually much faster.

6 Structure of generalized Ehrhart series Theorem E M;f.t/ D 1 C h 1t C C h s t s.1 t`/ rank MCdeg f ; h i 2 Q; s < `.rank M C deg f/: where ` is the lcm of deg x i, x i 2 Z d, i D 1; : : : ; m, generating R C M. Equivalently Theorem There exists a quasipolynomial q D q M;f 2 QŒX of degree g < rank M C deg f and period j ` such that E.P; f; k/ D q.k/; k 0: A quasipolynomial is a polynomial with periodic coefficients: q.k/ D c.k/ g k g C C c.k/ 1 k C c.k/ 0 ; c.k/ i D c.j/ i if j k./:

7 Lead coefficient and Lebesgue integral The virtual leading coefficient of the quasipolynomial is constant: Proposition c rank MCm 1 D Z P f m d; m D deg f; where f m is the leading homogeneous component of f, and is the Lebesgue masure under which a basic lattice cube in aff.p/ has volume 1. Often the leading coefficient is the most important information, since it controls the growth order of E.M; f; k/. NmzIntegrate also computes Lebesgue integrals of (not necessarily homogeneous) polynomials over rational polytopes.

8 Application: The Condorcet paradox A paradigmatic paradox of the theory of social choice has been named after the Marquis de Condorcet ( ). Consider an election with 3 candidates, called A,B,C. Each of the k fixes a preference order, in other words, a linear order of the candidates. There 6 such orders: P 1 P 2 P 3 P 4 P 5 P 6 A A B B C C B C A C A B C B C A B A Election result: x i of the k voters choose the preference order P i. Then x 1 C C x 6 D k, x i 2 Z, x i 0 for i D 1; : : : ; 6.

9 Who is the winner? Usually the majority winner, the candidate with most first places. But one could also take the Condorcet winner (CW): A beats B if the number of voters that prefer A to B is larger than the number of voters with the opposite preference. A is the Condorcet winner if he beats B and C. Observation of Condorcet (and others): there are election results for which beats is not transitive! In particular, there need not exist a CW. This is the Condorcet paradox. The standard assumption for the following quantification: a priori, all election results have equal probability.

10 Quantification A is the Condorcet winner if 1.x/ D x 1 C x 2 x 3 x 4 C x 5 x 6 > 0 2.x/ D x 1 C x 2 C x 3 x 4 x 5 x 6 > 0 If there are k voters, then the probability of A is the CW is p k.a CW/ D #fx W i.x/ > 0; i D 1; 2g #fall xg D #fx W i.x/ > 0; i D 1; 2g kc5 6 The probability that there is a CW at all then is p k.cw/ D 3 p k.a CW/ Obviously, for large k the limit as k! 1 is (more) interesting: p.cw/ D lim k!1 p k.cw/:

11 Election results as lattice points The potential election results for a fixed number of k voters are exactly the lattice points in the simplex S k D fx 2 R 6 W x i 0; X x i D kg und the results that have a CW are the lattice points in the subset C k D fx 2 S k W i.x/ > 0; i D 1; 2g Attention: C k is a semiopen polytope since the lattice points in some faces (in this case: faects) do not belong to C k : C k D C k n.union of some faces/ Normaliz and NmzIntegrate can now (current development versions) deal with semiopen polytopes and semiopen monoids.

12 Symmetries Once more the inequalities for A is the CW (3 candidates): 1.x/ D x 1 C x 2 x 3 x 4 C x 5 x 6 > 0 2.x/ D x 1 C x 2 C x 3 x 4 x 5 x 6 > 0 Achill Schürmann s observation: the inequalities are symmetric with respect to x 1 $ x 2 and x 4 $ x 6. Simplification: 1.y/ D y 1 y 2 y 3 C y 4 > 0 y 1 D x 1 C x 2 ; y 3 D x 3 2.y/ D y 1 y 2 C y 3 y 4 > 0 y 2 D x 4 C x 6 ; y 4 D x 5 Hence we consider the projection W R 6! R 4 and replace the polytope C by D D.C/. Gain: simpler geometry. Loss: more difficult counting problem. Computations for 3 candidates are very easy, but for 4 candifates they are already extremely hard without symmetrization.

13 The enormous gain of symmetrization The following table shows the partly extreme acceleration of computations by use of the symmetrization. dim deg f # triang/ # dec time E series lead coeff Cond paradox : / 1: sec 3.2 sec Cond. parad. symm / sec 0.07 sec Cond. eff. plurality : / 4: h Cond. eff. pl. symm ,953/ 23,453 3:34 h 25:53 min Plur. vs. cutoff : / 4: h Plur. vs. cut symm / sec 0.18 sec SUN xfire 4450, 20 parallel threads

14 Back to generalized Ehrhart functions We restrict ourselves to closed polytopes and monoids: the semiopen case is reduced to the closed case via inclusion/exclusion. Recall that we want to compute E M;f.t/ D X x2m f.x/t deg x D 1 C h 1t C C h s t s.1 t`/ rank MCdeg f for a monoid M D C \ Z d with grading deg. We proceed in 3 steps: 1 M D Z C : direct approach 2 M D Z d C : induction on d 3 the general case by decomposing M into suitable images of Z d (Stanley decomposition)

15 M D Z C By linearity, it is enough to consider P 1 kd0 k m t uk. Example: 1X 1X X kt k D.k C 1/t k t k 1 D 1 t 1 t D t.1 t/ : 2 kd0 kd0 kd0 So we write k m as a linear combination of the rising factorials.k C 1/ j D.k C 1/.k C 2/.k C j/; j D 0; : : : ; m (coefficients of k m : Stirling numbers of the second kind) and use 1X.j/ 1.k C 1/ j t k jš D D 1 t.1 t/ : jc1 kd0 After substitution t 7! t u : 1X k m t uk D A m.t u /.1 t u / mc1 (Eulerian numbers) kd0

16 M D Z d C If f.x/ D g.y/h.z/, y D.x 1 ; : : : ; x r /, z D.x rc1 ; : : : ; x d /, then E M;f.t/ D X X X f.x/t deg x D g.y/t deg y h.z/t deg z x2z d y2z r C C z2z d C In the general case we split off the last variable: r f.x/ D X i f i.x 1 ; : : : ; x d 1 /x i d ; and use E M;f.t/ D X i D X i with u D deg e d. X x 0 2Z d 1 C A i.t u / X.1 t u / ic1 X 1! f i.x 0 /t deg x0 k i t ui x 0 2Z d 1 C kd0 f i.x 0 /t deg x0!

17 Stanley decomposition = discrete triangulation Theorem (Stanley) Let M D C \ Z d, rank M D d. There exists a finite decomposition M D [ s s.z d C / (disjoint) where s W Z d C! M is a Z-linear affine map: s.x 1 ; : : : ; x d / D u.s/ C dx id1 x i v.s/ i : Consequence: E M;f.t/ D X X t deg u.s/ s x2z d C f. s.x//t deg s.x/ deg s.x/ D X x i deg v.s/ i

18 The crucial argument for Stanley decomposition Theorem Let be a triangulation of a polytope P. Then there exists a decomposition [ P D n S 2 ; dim Ddim P where S is a union of (at most dim P) facets of. Stanley s proof uses a line shelling (Bruggesser-Mani). Since version 2.7, Normaliz uses a theorem of Köppe-Verdoolaege.

19 Integration In the paper of Jeffries, Montaño and Varbaro on j-multiplicities and "-multiplicities we find the formula Z Y n m e.a t.x// D c.z 1 z m /.z j z i / 2 d PŒ0;1 m 1i<jm zdt This is an interesting example of an integral of a polynomial over a rational polytope. NmzIntegrate handles such integrals as follows: 1 triangulation of the polytope, 2 linear transformation from a rational simplex to the unit simplex, 3 integration over the unit simplex.

20 Integration over the unit simplex We consider the unit.d 1/-simplex naturally embedded into R d as the convex hull of the unit vectors. Integration over the unit simplex is done monomial by monomial: Proposition Z y m 1 1 y m d d d D m 1 Š m d Š.m 1 C C m d C d 1/Š :