INTERMEDIATE SUMS ON POLYHEDRA: COMPUTATION AND REAL EHRHART THEORY

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INTERMEDIATE SUMS ON POLYHEDRA:

COMPUTATION AND REAL EHRHART THEORY

V Baldoni, N Berline, Matthias Koeppe, M. Vergne

To cite this version:

V Baldoni, N Berline, Matthias Koeppe, M. Vergne. INTERMEDIATE SUMS ON POLYHEDRA:

COMPUTATION AND REAL EHRHART THEORY. 2014. �hal-01087290�

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COMPUTATION AND REAL EHRHART THEORY

V. BALDONI, N. BERLINE, M. K ¨OPPE, AND M. VERGNE

Abstract. We study intermediate sums, interpolating between integrals and discrete sums, which were introduced by A. Barvi- nok [Computing the Ehrhart quasi-polynomial of a rational simplex, Math. Comp. 75 (2006), 1449–1466]. For a given semi-rational polytopepand a rational subspaceL, we integrate a given polynomial functionhover all lattice slices of the polytopep parallel to the subspaceLand sum up the integrals. We first develop an al- gorithmic theory of parametric intermediate generating functions.

Then we study the Ehrhart theory of these intermediate sums, that is, the dependence of the result as a function of a dilation of the polytope. We provide an algorithm to compute the resulting Ehrhart quasi-polynomials in the form of explicit step polynomials. These formulas are naturally valid for real (not just integer) dilations and thus provide a direct approach to real Ehrhart theory.

Contents

1. Introduction 2

2. Notations and basic facts 4

2.1. Rational and semi-rational polyhedra 4

2.2. Valuations and generating functions 4

3. Intermediate generating function for a simplicial affine cone 7 3.1. Where the slicing space is parallel to a face 7 3.2. Cone decomposition with respect to a linear subspace 8 3.3. Complexity of the intermediate generating function 14 4. Ehrhart quasipolynomials for intermediate valuations 19

Acknowledgments 22

References 23

Date: November 27, 2010 (Revision 259).

2010 Mathematics Subject Classification: 05A15 (Primary); 52C07, 68R05, 68U05, 52B20 (Secondary).

1

arXiv:1011.6002v1 [math.CO] 27 Nov 2010

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1. Introduction

Let p be a rational polytope in V = R^d and h(x) a polynomial function on V. A classical problem is to compute the sum of values of h(x) over the set of integral points ofp,

S(p, h) = X

x∈p∩Z^d

h(x).

The sum S(p, h) has generalizations, the intermediate sums S^L(p, h), whereL⊆V is a rational vector subspace, introduced by Barvinok [6].

They interpolate between the discrete sum S(p, h) and the integral R

ph(x) dx. For a polytopep⊂V and a polynomialh(x) S^L(p, h) = X

x

Z

p∩(x+L)

h(y) dy,

where the summation index x runs over the projected lattice in V /L.

In other words, the polytopep is sliced along affine subspaces parallel to L through lattice points and the integrals of h over the slices are added up. For L =V, there is only one term and S^V(p, h) is just the integral ofh(y) over p, while, forL={0}, we recover the discrete sum S(p, h). As in the discrete case, a powerful method is to consider the intermediate generating function

S^L(p)(ξ) = X

x

Z

p∩(x+L)

e^hξ,yidy, (1)

forξ∈V^∗. Ifpis a polyhedron, not necessarily compact, the generating function still makes sense as a meromorphic function. By writing a polyhedron as the sum of its cones at vertices (Brion’s theorem), we need only study the case where p is an affine cone.

It is then natural to turn to the Ehrhart theory of the intermediate sums S^L, that is, the study of the intermediate sum S^L(tp, h) of a dilated polytopetpas a function of the dilation parametert. It turns out that, just like in the classical case, the Ehrhart function t7→S^L(tp, h) is a quasi-polynomial, that is, a function of the form

S^L(tp, h) =

d+M

X

m=0

E_m^L(p, h;t)t^m, (2) where the coefficients E_m^L(p, h;t) depend only on tmodq, where q is an integer such that qphas lattice vertices.

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The main results of this article are:

1. a polynomial time algorithm for the computation of the intermediate generating function of a simplicial affine cone, when the slicing space has fixed codimension, Theorem 24,

2. a polynomial time algorithm for the computation of the weighted intermediate sum S^L(p, h) of a simple polytopep (given by its vertices), and the corresponding Ehrhart quasipolynomialt7→S^L(tp, h), when the slicing space has fixed codimension and the weight depends only on a fixed number of variables, or has fixed degree, Theorem28.

The main new feature of this article is the use of a decomposition of any simplicial cone into simplicial cones which have a face parallel to the given space L, the decomposition being given by a closed formula, Theorem 18. This decomposition is borrowed from [9] where it plays a key role in the study of partition functions. We prove that this decomposition is done by a polynomial algorithm, if the codimension of L is fixed.

Thus, the computation of the intermediate generating function of a simplicial affine cone is reduced to the case whereLis parallel to a face of the cone, and thus the generating function factors into a discrete sum and an integral. This case has been already studied in [2]. The output of the algorithm is a “short formula” where both ξ and the vertex s of the cone appear as symbolic variables (see Examples 6, 7 and 26).

Once a short formula for the generating function is available, we follow the method of [2] for the computation of the weighted Ehrhart quasi- polynomial of a simple polytope. The method uses the fact that the intermediate generating function decomposes into meromorphic terms of homogeneous ξ-degree. The output is again a short formula where the dilating parameter t appears as a symbolic variable inside step polynomials, see Example 30.

The results remains valid for non-negative real values of t. Thus our method provides a direct proof of a real Ehrhart theorem (Theo- rem 31). This extension of Ehrhart theory to real dilation factors has been studied by E. Linke [13] for the classical case. It is even more natural for the intermediate valuations, as one of the cases is L = V. Then S^L(tp, h) is the integral of h over thetp, the result being clearly a polynomial formula valid for any non-negative real value of t.

The present manuscript together with the article [2] also provide the foundation for a future work, in which the weighted Barvinok patched sums P

Lλ(L)S^L(p, h) for a simple polytope will be studied.

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2. Notations and basic facts

2.1. Rational and semi-rational polyhedra. We consider a rational vector space V of dimension d, that is to say a finite dimensional real vector space with a lattice denoted by Λ. We will need to consider subspaces and quotient spaces of V, this is why we cannot just letV =R^d and Λ =Z^d. An element v ∈V is called rational if nv ∈Λ for some non-zero integern. The set of rational points inV is denoted byV_Q. A subspace LofV is calledrational ifL∩Λ is a lattice inL. If L is a rational subspace, the image of Λ inV /L is a lattice in V /L, so that V /L is a rational vector space. The image of Λ in V /L is called the projected lattice. It is denoted by Λ_{V /L}. A rational space V, with lattice Λ, has a canonical Lebesgue measure dx = dm_Λ(x), for which V /Λ has measure 1.

Aconvex polyhedron pinV (we will simply saypolyhedron) is, by definition, the intersection of a finite number of closed half spaces bounded by affine hyperplanes. If the hyperplanes are rational, we say that the polyhedron is rational. If the hyperplanes have rational directions, we say that the polyhedron is semi-rational. For instance, if p ⊂ V is a rational polyhedron, t is a real number and s is any point in V, then the dilated polyhedrontpand the translated polyhedrons+pare semi- rational. Unless stated otherwise, a polyhedron will be assumed to be semi-rational.

In this article, a cone is a polyhedral cone (with vertex 0) and an affine cone is a translated sets+cof a conec. A cone cis calledsimplicial if it is generated by independent elements ofV. A simplicial conec is called unimodular if it is generated by independent integral vectors v₁, . . . , v_k such that {v₁, . . . , v_k} can be completed to an integral basis ofV. An affine coneais calledsimplicial (resp.simplicial unimodular) if the associated cone is.

A polytope p is a compact polyhedron. The set of vertices of p is denoted by V(p). For each vertexs, the cone of feasible directions at s is denoted bycs.

We denote by L(V) the space of functions on V which is generated by indicator functions [q] of polyhedra qwhich contain lines.

2.2. Valuations and generating functions.

Definition 1. Let M be a vector space. An M-valued valuation Φ is a map from the set of polyhedra p⊂V to the vector spaceM such that whenever the indicator functions [p_i]of a family of polyhedra p_i satisfy a linear relationP

ir_i[p_i] = 0, then the elementsΦ(p_i)satisfy the same

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relation

X

i

r_iΦ(p_i) = 0.

Definition 2. We denote by H(V^∗) the ring of holomorphic functions defined around 0∈V^∗. We denote byM(V^∗)the ring of meromorphic functions defined around0∈V^∗ and by M_`(V^∗)⊂ M(V^∗)the subring consisting of meromorphic functions φ(ξ) which can be written as a quotient of a holomorphic function by a product of linear forms.

The intermediate generating functions of polyhedra which are studied here are important examples of functions inM_`(V^∗).

Proposition 3. Let L ⊆ V be a rational subspace. There exists a unique valuationS^L which to every semi-rational polyhedronp⊂V as- sociates a meromorphic function S^L(p)∈ M(V^∗) so that the following properties hold:

(a) If p contains a line, then S^L(p) = 0.

(b)

S^L(p)(ξ) = X

x∈Λ_{V /L}

Z

p∩(x+L)

e^hξ,yidm_L(y), (3) for every ξ ∈V^∗ such that the above sum converges.

(c) For every point s ∈Λ +L, we have

S^L(s+p)(ξ) = e^hξ,siS^L(p)(ξ).

When necessary we will indicate the lattice and use the notation S^L(p,Λ).

Remark 4. The fact that S^L(p)is actually an element of M_`(V^∗)will be a consequence of the explicit computations of the next section.

ForL={0}, we recover the valuationS given by S(p)(ξ) = X

x∈p∩Λ

e^hξ,si,

provided this sum is convergent. For L=V, S^V(p)(ξ) is the integral I(p)(ξ) =

Z

p

e^hξ,xidx,

if p is full dimensional, and S^V(p) = 0 otherwise. If p is compact, the meromorphic function S^L(p)(ξ) is actually regular at ξ = 0, and its value for ξ = 0 is the valuation E_L^⊥(p) considered by Barvinok [6].

The proof is entirely analogous to the cases L = {0} and L = V, see for instance Theorem 3.1 in [7] and we omit it.

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Remark 5 (Rational vs. semi-rational polyhedra). Although [7] and other classical texts deal only with rational polyhedra, their proofs extend easily to semi-rational ones. For instance, by summing geometric series, one obtains immediately the generating function of a unimodular cone with a real vertex.

Here are some simple examples. For t ∈ R we denote by {t} the fractional part of t, i.e., 0 ≤ {t} < 1 and t− {t} ∈ Z, and by dte the ceiling of t, i.e., dte=t+{−t}.

Example 6. For anyt∈R, we have S(t+R≥0)(ξ) = e^dteξ

1−e^ξ = e^tξe^{−t}ξ 1−e^ξ.

More generally, letc⊂R^d be a unimodular cone with primitive edge gener- atorsv₁, . . . , v_d. Letsbe any point inR^d, withs=Pd

i=1s_iv_i. Then S(s+c)(ξ) =

d

Y

i=1

e^dsⁱ^ehξ,vⁱⁱ

1−e^hξ,vⁱⁱ = e^hs,ξi

d

Y

i=1

e^{−sⁱ^}hξ,vⁱⁱ 1−e^hξ,vⁱⁱ.

Example 7. Let c ⊆ R² be the first quadrant and L = Re1. We write s= (a, b)∈R². We have

S^L(s+c)(ξ1, ξ2) =

∞

X

n=dbe

Z +∞

a

e^nξ²^+xξ¹dx= ∞

X

n=dbe

e^nξ²

·−e^aξ¹ ξ₁

= e^dbeξ²

1−e^ξ² ·−e^aξ¹ ξ1

= e^hs,ξie^{−b}ξ² 1−e^ξ² ·−1

ξ1

. Example 8. Let p ⊂ R² be the triangle with vertices (0,0), (1,0), (0,1).

LetL=R(1,0). Straightforward calculations giveS^L(p) = ^1−e_ξ^ξ¹

1 ,I(p)(ξ) =

1 ξ1−ξ2

1−e^ξ1

ξ1 −^1−e_ξ^ξ²

2

, andS(p)(ξ) = 1 + e^ξ¹+ e^ξ². These three functions are indeed analytic.

A consequence of the valuation property is the following fundamental theorem. It follows from the Brion–Lawrence–Varchenko decomposition of a polytope into the supporting cones at its vertices [8].

Theorem 9 (Brion’s theorem). Let p be a polytope with set of vertices V(p). For each vertex s, let c_s be the cone of feasible directions at s.

Let L⊆V be a rational subspace. Then S^L(p) = X

s∈V(p)

S^L(s+c_s).

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v₂

v₁

Figure 1. The projected lattice Λ{1}

3. Intermediate generating function for a simplicial affine cone

3.1. Where the slicing space is parallel to a face. Let c ⊂ V be a simplicial cone with integral generators (v_j, j = 1, . . . , d), and lets ∈V. Let us recall the expression for the intermediate generating functionS^L(s+c)(ξ), when the slicing subspaceLis generated by some of the v_j’s.

For I ⊆ {1, . . . , d}, we denote by L_I the linear span of the vectors (v_i, i ∈ I). We denote the complement of I in {1, . . . , d} by I^c. We haveV =L_I⊕L_I^c. For x∈V we denote the components by

x=x_I +x_I^c.

Thus we identify the quotient V /L_I with L_I^c and we denote the projected lattice by Λ_I^c ⊂L_I^c. Note thatL_I^c∩Λ⊆Λ_I^c, but the inclusion is strict in general.

Example 10. Letv1 = (1,0),v2 = (1,2),I ={2}. The projected lattice Λ_I^c on L_I^c =Rv₁ isZ^v₂¹. See Figure 1.

We denote by c_I the cone generated by the vectorsv_j, for j ∈I and by b_I the parallelepiped b_I =P

i∈I[0,1[v_i. Similarly we denote by c_I^c the cone generated by the vectorsv_j, forj ∈I^candb_I^c =P

i∈I^c[0,1[v_i. The projection of the cone c on V /L_I = L_I^c identifies with c_I^c. Note that a generator v_i, i ∈ I^c, may be non-primitive for the projected lattice ΛI^c, even if it is primitive for Λ, as we see in the previous example. We write s=sI+sI^c.

First let us recall from [2] how the intermediate generating function S^L^I(s+c,Λ) decomposes as a product. This already appeared in Example 7.

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Proposition 11. [2]. The intermediate sum for the full cone s +c breaks up into the product

S^L^I(s+c,Λ)(ξ) =S(s_I^c+c_I^c,Λ_I^c)(ξ)I(s_I+c_I, L_I∩Λ)(ξ). (4) We remark that in the above equation, the functionS(s_I^c+c_I^c,Λ_I^c)(ξ) is a meromorphic function on the space (L_I^c)^∗. The integral I(s_I + c_I, L_I∩Λ)(ξ) is a meromorphic function on the space (L_I)^∗. We consider both as functions onV^∗through the decompositionV =L_I⊕L_I^c. 3.2. Cone decomposition with respect to a linear subspace.

Let L be a linear subspace of V. Let c be a full dimensional simplicial cone in V with generators w1, . . . , wd. The main theorem of this section, Theorem 18, gives a signed decomposition ofcinto a family of full dimensional simplicial cones cσ, where each cσ has a face parallel to L. More precisely, this decomposition is a relation between indicator functions of cones, modulo the spaceL(V) of functions onV which is generated by indicator functions [q] of polyhedra q which contain lines. This decomposition is borrowed from [9]. The proof relies on the following simple lemma which has its own interest. The extreme cases, dimL= 1 or codimL= 1, of Theorem 18follow easily from this lemma (Proposition 14). The general case is proven by induction on either dimL or codimL.

Given vectors u₁, . . . , u_p ∈ R^d, we denote the cone Pp

i=1R^≥0u_i by c(u₁, . . . , u_p) and its relative interior Pp

i=1R>0u_i by ˜c(u₁, . . . , u_p).

Lemma 12. Let u₁, . . . , u_d be a basis of V and let u_d+1 =−Pd k=1u_k. For i= 1, . . . , d+ 1, define semi-open cones

k_i =c(u₁, . . . , ui−1) + ˜c(u_i+1, . . . , u_d+1).

Then V is the disjoint union of the k_i for i= 1, . . . , d+ 1.

Proof. Let x = t₁u₁ +. . .+t_du_d. If t_k ≥ 0 for all k, then x ∈ k_d+1. Otherwise, let i be the largest index with 1 ≤ i ≤ d, for which the minimum min1≤k≤dt_k is reached. Writing u_i = −P

1≤k≤d+1,k6=iu_k, we obtain

x= X

1≤k≤d,k6=i

(tk−ti)uk−tiud+1.

As tk−ti ≥0 if 1≤j ≤d and tj−ti >0 if i < j ≤d, and ti <0, we see thatx∈ki.

Let us show that ki ∩ kj is empty if i < j. Let x ∈ ki, x = P

1≤k≤d+1,k6=it_ku_k. Writing u_j =−P

1≤k≤d+1,k6=ju_k, we obtain

x= X

1≤k≤d, k /∈{i,j}

(t_k−t_j)u_k−t_ju_i.

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As tj >0, it follows that x /∈ki. From the lemma, we deduce other interesting cone decompositions.

The first one is the well-known stellar decomposition.

Proposition 13 (Stellar decomposition). Letw₁, . . . , w_d be a basis of V. Let r ≤s and

v =w1+· · ·+wr−(ws+1+· · ·+wd).

Let c be the cone generated by w_i’s for 1≤i ≤d and let k be the cone generated by c and the vector v.

(a) Define the semi-open cones k_i by

k_i = ˜c(w₁, . . . , wi−1) +c(w_i+1, . . . , w_d, v) for 1≤i≤r, if r≥1, k_i =c(w₁, . . . , wi−1) + ˜c(w_i+1, . . . , w_d, v) for s+ 1≤i≤d, if s < d.

Then:

(a.1) If r≥1, k is the disjoint union of the k_i for 1≤i≤r.

(a.2) k\c is the disjoint union of the k_i for s+ 1≤i≤d. (If s=d, then k\c=∅.)

(b) For i∈ {1, . . . , r} ∪ {s+ 1, . . . , d}, let

c_i =c(w₁, . . . , w_i−1, w_i+1, . . . , w_d, v).

Then

[c]−

r

X

i=1

[c_i] +

d

X

i=s+1

[c_i]≡0

modulo indicator functions of lower dimensional cones.

Proof. Let us prove (a.2) first. By quotienting out by the subspace generated by w₁, . . . , w_s, we can assume that s = 0, thus v = −(w₁+ . . .+w_d). In this case, we just apply Lemma 12 with u_k = w_k and u_d+1=v. As the cone k_d+1 of Lemma 12 is justc, we obtain (a.2).

Let us now assume that l ≥1. We write

w₁ =v+w_d+· · ·+w_s+1−(w_r+· · ·+w₂).

We apply (a.2) with ¯w₁ =v, ¯w₂ =w_detc., and ¯v =w₁. The corresponding semi-open cones are ¯k₁ =c(v, w_d, . . . , w_r+1) + ˜c(wr−1, . . . , w₂, w₁) = kr, . . . ,¯kr =c(v, wd, . . . , w2) = k1, thus we obtain (a.1).

For i ∈ {1, . . . , r} ∪ {s+ 1, . . . , d}, the (closed) cone ci differs from the semi-open cone k_i by the union of some faces of lower dimension, therefore (b) follows from (a) and inclusion-exclusion relations.

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Next, we obtain decompositions of any simplicial cone into full- dimensional cones which have a given edge, or a given facet. Recall that L(V) is the linear span of indicator functions of cones which contain straight lines.

Proposition 14. Let w₁, . . . , w_d be a basis of V and let c be the cone generated by w_i’s for 1≤i≤d. Let 0≤r ≤s≤d+ 1.

a) Let

v =w₁+· · ·+w_r−(w_s+1+· · ·+w_d). (5) Then we have the following relation between indicator functions of cones.

[c]≡

r

X

i=1

(−1)ⁱ⁻¹[c(−w₁, . . . ,−wi−1, w_i+1, . . . , w_d, v)]

+

d

X

i=s+1

(−1)^d−i[c(w₁, . . . , wi−1,−w_i+1, . . . ,−w_d,−v)] mod L(V).

(6) b) Let L ⊂ V be a hyperplane. Assume that w_i ∈ L if and only if r+ 1≤i ≤s, that the w_i’s lie on one side of L for 1≤ i≤ r and on the other side for s+ 1 ≤ i ≤ d. For i ∈ {1, . . . , r} ∪ {s+ 1, . . . , d}, denote by ρi the projection V → L parallel to wi. Then we have the following relation between indicator functions of cones.

[c]≡

r

X

i=1

[R⁺wi+ρi(c)]−

d

X

i=s+1

[R⁺(−wi) +ρi(c)] mod L(V). (7) Example 15. c=c(w1, w2),v=w1+w2. Then

[c] =−[c(−w₂, v)] + [c(w₁, v)] + [c(w₂,−w₂, w₁)].

Indeed [c(w2,−w₂, w1)]−[c] is the indicator function of the quadrantc(w1,−w₂) minus that of the lineRw1. This is also the case for [c(−w₂, v)] + [c(w1, v)].

≡

a L

w1modL σ={1}

cσ

w2

ρ{1}(w2) {1},1w1

−

a L

cσ

σ={2}

w2modL

{2},2w2

w2

ρ{2}(w1)

Figure 2. Brion–Vergne decomposition of a conecinto cones with a face parallel to the subspace L, modulo cones with lines. The vectors in the quotientV /Ldeter- mine the signs _σ,j.

Theorem 18 (Brion–Vergne decomposition, [9], Theorem 1.2). Let L be a linear subspace of V. Let c be a full dimensional simplicial cone in V with generators w₁, . . . , w_d. Fix a vector a ∈V /L that is generic with respect to the pair (c, L) and belongs to the projection ofc onV /L.

For σ∈ B(c, L), let

a =X

j∈σ

a_σ,j(w_j modL).

Let _σ,j be the sign of a_σ,j and (σ) =Q

j∈σ_σ,j. Denote by c_σ ⊂V the cone with edge generators

_σ,jw_j for j ∈σ, and ρ_σ(w_k) for k /∈σ.

Then we have the following relation between indicator functions of cones.

[c]≡ X

σ∈B(c,L)

(σ) [c_σ] mod L(V). (8)

The theorem is illustrated in Figure 2.

Remark 19. As explained in [9], the case where dimL= 1 reduces to case a) of Proposition 14. Let us show that the case wherecodimL= 1 reduces to case b). A vectora inV /L is generic with respect to(c, L)if and only if a 6= 0. Up to renumbering, we can assume that w₁, . . . , w_r lie on the same side of Las a, while w_s+1, . . . , w_d lie on the other side.

The set B(c, L) consists of the singletons σ_i = {i} such that w_i ∈/ L, ie i∈ {1, . . . , r} ∪ {s+ 1, . . . , d}. The sign _σ_i_,i is +1 if i∈ {1, . . . , r}

and −1 if i ∈ {s+ 1, . . . , d}, therefore Equation (8) becomes precisely Equation (7).

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Theorem 20 (Efficient construction of a Brion–Vergne decomposition). Fix a number k₀. There exists a polynomial-time algorithm for the following problem. Given a rational simplicial cone c and a rational subspace L of V of codimension at most k₀, compute a decomposition (8).

Proof. By Remark 17, we can enumerate the bases σ ∈ B(c, L) in polynomial time.

A rational generic vector a can be constructed in polynomial time as follows; this technique was already used in [5]. Fix one of the bases

¯

σ ∈ B(c, L) as a basis of V /L. Then the projections ofw₁, . . . , w_dhave rational coordinate vectors ¯w₁, . . . ,w¯_d∈Q^k⁰ with respect to this basis.

Consider the moment curve ¯a(t) = (1, t, t², . . . , t^k⁰⁻¹) ∈R^k+⁰ for t ≥ 0. For any σ ∈ B(c, L) and j ∈ σ, we have by Cramer’s rule that the coefficient aσ,j(t) for a = a(t) is nonzero if and only if fσ,j(t) = det(a(t); ¯wi : i ∈ σ, i 6= j) 6= 0. The functions fσ,j(t) are polynomial functions in t of degree at most k₀(k₀ −1), which are not identically zero. Hence each has at most k₀(k₀ −1) zeros. Thus, by computing the coefficients a_σ,j(t) for M =k₀(k₀ −1)|B(c, L)|+ 1 different values for t, we will find one t = ¯t that is not a zero of any f_σ,j. Clearly this search can be implemented in polynomial time.

Then the vectora=P

i∈σ¯a_i(t)(w_i modL)∈V /Lis generic with respect to the pair (c, L), and since ¯a_i(t)≥0, it also lies in the projection of conV /L. (We could also use explicit deterministic constructions of generic vectors of polynomial encoding size, without search, similar to

[11, 12].)

Remark 21(Relation to variants of Barvinok’s decomposition).Barvi- nok’s method to efficiently compute the discrete generating function S(s+c)(ξ) of a cone in fixed dimension decomposes a simplicial cone into cones with smaller indices (and ultimately into unimodular cones).

The original method by Barvinok [5] used a primal decomposition and inclusion-exclusion to take care of lower-dimensional cones. The dual of the stellar decomposition is used in the variant of Barvinok’s method that was presented in [7] and later implemented in LattE [10]; this avoids having to take care of lower dimensional cones. Instead of du- alizing, one can directly uses Formula (6), the case where dimL = 1.

We use this primal variant of Barvinok’s decomposition in our imple- mentation. Other primal variants were studied in [11] and [12] and implemented in LattE macchiato.

3.3. Complexity of the intermediate generating function. We will use the following notations.

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Definition 22. Let

T(t, x) = e^tx x

1−e^x =−

∞

X

n=0

B_n(t)xⁿ

n!, (9)

where B_n(t) are the Bernoulli polynomials.

Definition 23. For a and q a positive integer, we denote bnc_q :=q₁

qn

∈qZ, {n}_q := (nmodq)∈[0, q), which give the unique decomposition

n =bnc_q+{n}_q.

Thus bac₁ = bac and {a}₁ = {a} will be the ordinary “floor” and

“fractional part” of a.

Theorem 24 (Short formula for S^L(s +c)(ξ)). Fix a non-negative integer k₀. There exists a polynomial time algorithm for the following problem. Given the following input:

(I₁) a numberd in unary encoding,

(I₂) a simplicial cone c = c(v₁, . . . , v_d) ⊂ R^d, represented by the primitive vectors v₁, . . . , v_d∈Z^d in binary encoding,

(I₃) a rational subspace L ⊆ R^d of codimension k₀, represented by d−k₀ linearly independent vectors b₁, . . . , bd−k0 ∈ Z^d in binary encoding,

compute the following output in binary encoding:

(O₁) a finite set Γ,

(O₂) for every γ in Γ, integers α^(γ) and lattice vectors w^(γ)_i ∈Z^d for i= 1, . . . , d,

which have the following properties.

(1) For every γ, the family(w_i^(γ)), for i= 1, . . . , d, is a basis of R^d and if we denote the dual basis by(η_i^(γ)), fori= 1, . . . , d, thenη_i^(γ) ∈Z^d for i= 1, . . . , k₀.

(2) For every s ∈ R^d, we have the following equality of meromorphic functions of ξ:

S^L(s+c,Z^d)(ξ)

= e^hξ,siX

γ∈Γ

α^(γ)

k0

Y

i=1

T

−

η_i^(γ), s ,

ξ, w^(γ)_i

· 1

Qd i=1

ξ, w_i^(γ). (10)

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Remark 25. Consider the term corresponding toγ ∈Γin (10). Define s^(γ)=

k0

X

i=1

−

η_i^(γ), s w^(γ)_i .

The point s+s^(γ) is inLk0

i=1Zw^(γ)_i ⊕Ld

i=k0+1Rw^(γ)_i , and formula (10) reads also

S^L(s+c,Z^d)(ξ) = X

γ∈Γ

α^(γ)e^hξ,s+s^(γ)ⁱ 1 Qk0

i=1 1−e^hξ,wⁱ^(γ)ⁱ

1 Qd

i=k0+1hξ, w^(γ)_i i. (11) Formula (11) is easier to grasp, while (10) is used in the program, where we avoid all the singular hyperplanes hξ, w_i^(γ)i = 0, for γ ∈ Γ and i= 1, . . . , d, by deforming the linear form ξ into ξ+.

Proof. Using Theorem 18, we construct a signed decomposition of c into conesc^σ which have a face parallel toL, so that for anys ∈R^d we have

[s+c]≡X

σ

^σ[s+c^σ] (modulo cones containing lines), (12) where ^σ ∈ {±1}. Here σ runs through the set B(c, L) of Theorem 18, which can be enumerated in polynomial time by Remark 17. This makes the construction of the signed decomposition a polynomial-time algorithm. As the valuationS^L vanishes on cones which contain lines, we get

S^L(s+c)(ξ) = X

σ∈B(c,L)

^σS^L(s+c^σ)(ξ).

Each conec^σ is given by its edge generatorsw_i^σ ,i= 1, . . . , d, which are such that (w_i^σ), for i =k₀+ 1, . . . , d), is a basis of L. Then we apply Theorem 28 [Short formula for S^L^I(s+c,Z^d)(ξ)] of [2] with input c^σ and I ={k₀+ 1, . . . , d}. See Examples 6and 7.

Example 26 (Continuation of Example16). Again c is the cone c((0,−2,0),(1,−2,0),(0,−2,3)).

Here is the output of our Maple program computing the generating functions S^L(s+c)(ξ) (Formula (11)). We sets= (a, b, c) and ξ= (x, y, z).

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i) Let L be the subspace with basis (0,1,1). In this case, the three cones which occur in the decomposition of Example 16have non-unimodular projections in V /L, so that the projections are further decomposed into unimodular cones. This is why we have six terms in the expression ofS^L(s+c):

S^L(s+c)(x, y, z)

= e^ax+by+cz

e^{3a−b+c}(²⁵^y−³⁵z)+{a}(−x+3z)

(1−e²⁵^y−³⁵^z)(1−e^−x+3^z)(y+z) + e^{a}(−x+2^y)+{2^a+b−c}(−²⁵^y+³⁵^z)

(1−e^−x+2^y)(1−e⁻²⁵^y+³⁵^z)(y+z)

− e^−{3a−b+c}z+{−a}(x−3z)

(1−e^−z)(1−e^x−3^z)(−y−z) + e−{−b+c}z+{−a}x

(1−e^−z)(1−e^x)(−y−z) + e{−b+c}y−{a}x

(1−e^y)(1−e^−x)(−y−z) + e{a}(−x+2y)−{2a+b−c}y

(1−e^−x+2^y)(1−e^−y)(−y−z)

. ii) LetL be the subspace with basis (1,0,0), (0,1,1). Then:

S^L(s+c)(x, y, z)

= e^ax+by+cz

−6 e^{−b+c}(⁻¹₂^x+y)

1−e⁻¹²^x+y

x(−5x+ 6y+ 6z) + 6 e^{−b+c}(¹₃^x−z)

1−e¹³^x−z

x(−5x+ 6y+ 6z)

.

Example 27. We consider the four triangles Ti of Figure 3 which sub- divide the square q = [0,4]×[0,4]. Let L = R(0,1). Let us apply the inclusion-exclusion formula together with the valuation property of S^L. As S^Lvanishes on the edges and vertices which occur in the intersections of the triangles, it follows that the sumP4

i=1S^L(T_i) must be equal to S^L(q).

The example has been chosen so that the intermediate valuation S^L(q) can also be computed directly by elementary means. This allows us to verify our methods on the example. A direct computation gives

S^L(q) = ^e^4y_y⁻¹(1 + e^x+ e^2x+ e^3x+ e^4x).

Table1 shows the output of our Maple program that implements the algorithm of the theorem for S^L(Ti), i= 1, . . . ,4. The equalityP4

i=1S^L(Ti) = S^L(q) is readily checked on these expressions.

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Figure 3. The squareq subdivided into 4 triangles T_i. Lattice subspaces parallel to the subspace L = R(0,1) intersect the walls between the triangles at most in a point, and thus S^L vanishes on the intersections of the triangles.

Table 1. Intermediate valuations of the square and the 4 triangles

p VerticesV(p) Intermediate valuation S^L(p) T₁ (⁴₀),₅

2

4

,₀

11 3

e⁴^x 1−e^{−x+ 8}3y

y

− ^e⁴^x 1−e^{−x+ 11}12y

y + ^e³^{x+ 8}³^y 1−e^x−⁸3y

y

− ^e³^{x+ 61}¹⁵^y 1−e^{x+ 2}¹⁵^y

y

− ^e

113 y

1−e^x−¹¹¹²^y

y + ^e

113 y

1−e^{x+ 2}¹⁵^y y

T2 (⁰₀),(⁴₀), ₀

11 3

−_(1−e¹x)y − _(1−e^e⁴−x^x )y + ^e⁴^x 1−e^{−x+ 11}¹²^y

y + ^e

11 3 y

1−e^x−¹¹¹²^y y

T₃ ₅

2

4

,(⁴₀),(⁴₄) − ^e³^{x+ 8}³^y 1−e^x−⁸3y

y +_(1−e^e³^x+4x)y^y − ^e⁴^x 1−e^{−x+ 8}3y

y + _(1−e^e⁴^x+4−x^y)y

T4

₅

2

4

,(⁰₄),

₀

11 3

e²^x+4^y

(1−e^−x)y − ^e²^{x+ 59}¹⁵^y 1−e^−x−¹⁵²^y

y +_(1−e^e⁴^yx)y − ^e

113 y

(^1−e^x+2/15y)^y q (⁰₀),(⁰₄),(⁴₄),(⁴₀) ^e^4y_y⁻¹(1 + e^x+ e^2x+ e^3x+ e^4x)

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4. Ehrhart quasipolynomials for intermediate valuations Let us fix a rational polytope p and a weighth(x). It is well known (“Ehrhart’s Theorem”) that when we dilate p by an integer t ∈ N, the function t 7→S(tp, h) is a quasi-polynomial function of t of degree d+M where M is the degree ofh,

S(tp, h) =

d+M

X

m=0

E_m(p, h;t)t^m. (13) The coefficients E_m(p, h;t) depend only on {t}_q where q is an integer such thatqphas lattice vertices. Whenpis a lattice polytope, then the coefficients E_m(p, h;t) do not depend on t, and t 7→ S(tp, h) becomes a polynomial.

E. Linke [13] has proved that the expression (13) is still valid forreal dilations t ∈R. This is the generalized setting that we are going to use in this section. Note that if we allow real dilations, then even for lattice polytopespwe obtain a quasi-polynomial rather than a polynomial; the fractional part function {·}_q with q= 1 appears in the expressions for the coefficients E_m(p, h;t). For instance, the number of integers in the interval [0, t] =t[0,1] is t+ 1− {t}₁.

We are going to extend Ehrhart’s Theorem to the intermediate valuation S^L(p). Indeed, one still has

S^L(tp, h) =

d+M

X

m=0

E_m^L(p, h;t)t^m (14) where the coefficientsE_m^L(p, h;t) depend only on{t}_q. Note that when L = V, then S^L(tp, h) is polynomial, and coincides with the integral of h ontp.

More precisely, we will show that the Ehrhart coefficients are step- polynomial functions of t, in the sense of [14]. In the next theorem, we prove these results and furthermore, we show that the coefficients E_m^L(p, h;t) are given by short formulas, provided that in the input (L,p, h), the subspaceL has fixed codimension, p is a simple polytope given by its vertices, and the weight is a power of a linear form h(x) = h`, xi^M.

Theorem 28. For every fixed numberk₀ ∈N, there exists a polynomial- time algorithm for the following problem.

Input:

(I₁) a numberd∈N in unary encoding, with d≥k,

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(I2) a rational subspace L ⊆ R^d of codimension k0, represented by d−k₀ linearly independent vectors b₁, . . . , b_d−k₀ ∈Q^d in binary encoding,

(I₃) a finite index set V,

(I₄) a simple rational polytope p, given by the set V of its vertices, rational vectors s_i ∈Q^d in binary encoding,

(I₅) a rational vector `∈Q^d in binary encoding, (I₆) a numberM ∈N in unary encoding.

Output:

(O₁) a finite index set Γ,

(O₂) polynomialsf^γ,m ∈Q[r₁, . . . , r_k₀]and numbersζ_i^γ,m ∈Z, q^γ,m_i ∈ N for γ ∈Γ, i= 1, . . . , k₀ and m= 1, . . . , d+M,

such that for t ∈ R we have S^L(tp,h`, xi^M) = Pd+M

m=0 E_m(t)t^m, where E_m(t) is given by the step-polynomial

E_m(t) = X

γ∈Γ

f^γ,m {ζ₁^γ,mt}_q^γ,m

1 , . . . ,{ζ_k^γ,m

0 t}_q^γ,m

k0

.

Remark 29. Similar results hold for a more general weight h(x), as in [3] for integrals over a simplex. One can assume that the weight is given as a polynomial in a fixed number D of linear forms, h(x) = f(h`₁, xi, . . . ,h`_D, xi), or has a fixed degree D.

Proof of Theorem 28. As first observed in [4], the path from generating functions to Ehrhart quasi-polynomials relies on the following important property of functionsφ(ξ)∈ M_`(V^∗): such a function has a unique expansion into homogeneous rational functions

φ(ξ) = X

m≥m₀

φ_[m](ξ).

IfP is a homogeneous polynomial onV^∗ of degree p, andD a product of r linear forms, then ^P_D is an element in M_`(V^∗) homogeneous of degreem=p−r. For instance, ^ξ_ξ¹

2 is homogeneous of degree 0. On this example, we observe that a function in M`(V^∗) which has no negative degree terms need not be analytic.

For a fixed t ∈ R, the polynomial function ξ 7→S^L(tp,^hξ,xi_M!^M) is the term of degree M of of the holomorphic function S^L(tp)(ξ). Let c_s be the supporting cone of p at the vertex s. By Brion’s theorem applied to the semi-rational polytope tp, we writeS^L(tp)(ξ) as the sum of the intermediate generating functions of the cones at the verticests,s ∈ V, of the dilated polytope tp. The crucial point is that the cone c_s does

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Table 2. Intermediate Ehrhart quasi-polynomials of the square and the 4 triangles

p Ehrhart quasi-polynomialS^L(tp,1) T1 21

4 t²−⁷₈ ({4t}₁)²−₁₀⁷ {−5t}₂+⁷₈{4t}₁+₂₀⁷ ({−5t}₂)² T2 22

3 t²+¹¹₆ t− ¹¹₂₄ ({4t}₁)²+¹¹₂₄{4t}₁ T3 3t²+ 2−4{4t}₁

t− ¹₃({−5t}₂)²+²₃{−5t}₂+⁴₃({4t}₁)²−⁴₃{4t}₁ T₄ ₁₂⁵ t²+¹₆t−₆₀¹ ({5t}₂)²+₃₀¹ {5t}₂

q 16t²+ 4−4{4t}₁ t

not change when the polytope is dilated.

S^L(tp)(ξ) = X

s∈V

S^L(ts+cs)(ξ). (15) Let q_s be the smallest integer such that q_ss is a lattice point. Then (t− {t}_q_s)s is a lattice point, therefore we have

S^L(ts+c_s)(ξ) = e^hξ,(t−{t}^qs^)siS^L({t}_q_ss+c_s)(ξ)

= e^thξ,sie^−hξ,{t}^qs^siS^L({t}_q_ss+c_s)(ξ).

Expanding e^thξ,si in powers of t and looking for the coefficient of tⁱ, we obtain

S^L tp,^hξ,xi_M_!^M

=X

i≥0

tⁱX

s∈V hξ,siⁱ

i! e^−hξ,{t}^qs^siS^L({t}_q_ss+c_s)(ξ)

[M−i].

Thus we must analyze the term of ξ-degree M −i of e^−hξ,{t}^qs^siS^L({t}_q_ss+c_s)(ξ).

We use Formula (10) and consider the summands indexed by γ ∈ Γ, one at a time. From then on, the proof is entirely similar to the proof

of Theorem 37 in [2], and we omit it.

Example 30 (Example 27, continued). Table 2 shows the output of our Maple program for the Ehrhart quasi-polynomials of the four triangles of Example 27 with respect to the weight h = 1. Note that the dilating pa- rametertis real. We obtain