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Lattice sums of hyperplane arrangements

We introduce certain lattice sums associated with hyperplane arrangements, which are (multiple) sums running over integers, and can be regarded as generalizations of certain linear combinations of zeta-functions of root systems. We also introduce generating functions of special values of those lattice sums, and study their properties by virtue of the theory of convex polytopes. Consequently we evaluate special values of those lattice sums, especially certain special values of zeta-functions of root systems and their affine analogues. In some special cases it is possible to treat sums running over positive integers, which may be regarded as zeta-functions associated with hyperplane arrangements.


Introduction
The notion of Witten zeta-functions associated with semisimple Lie algebras was introduced by Zagier [12], inspired by the work of Witten [11] in quantum gauge theory. Recently the authors have developed the theory of zeta-functions of root systems (e.g. [4,5,6,7]), which are multi-variable generalizations of Witten zeta-functions. In particular, the "Weyl group symmetric" linear combinations of zeta-functions of root systems S(s, y; ∆) (where s is a complex multi-variable, y is a certain vector and ∆ is a finite reduced root system) and the generating functions of special values of those linear combinations were introduced and studied in [4,6,7].
In the present paper, we will introduce certain lattice sums of hyperplane arrangements, which are generalizations of the above linear combinations of zeta-functions of root systems. We will also introduce the generating functions of special values of those lattice sums. It is to be stressed that those generating functions can describe not only values but also functional relations among zeta-functions of root systems. Furthermore if they are combined with Poincaré polynomials of Weyl groups, we obtain explicit formulas for special odd values of zeta-functions of root systems. These results will be treated in the forthcoming paper [8].
Another application is to calculate special values of affine analogue of zeta-functions of root systems. Although in the cases of affine root systems it is natural to work with the character formulas instead of the dimension formulas, a straightforward generalization is also interesting. We will present some examples in Section 3.
In the present paper, our consideration is not restricted to the case in the domain of absolute convergence; we will study the values of lattice sums outside the domain of absolute convergence. Here we explain this point by simple examples.
Let N be the set of positive integers, N 0 = N ∪ {0}, Z the ring of rational integers, R the field of real numbers, and C the field of complex numbers. For any set S, the symbol S denotes the cardinality of S.
Let k ∈ N, and let y ∈ R with y / ∈ Z if k = 1. It is well-known (cf. [1,Theorem 12.19]) that where {y} = y − [y] is the fractional part of y and B k (·) is the k-th Bernoulli polynomial defined by In the case k = 0, then (1.1) does not hold straightforward. However this formula still holds in some sense via the following regularization. We see that the right-hand side of (1.1) is analytically continued to the whole space C in the variable k and then it is evaluated as −1 at k = 0. This effect is (formally) realized in the series by replacing the condition m = 0 by m = 0 and m =0 by − m=0 with 0 0 = 1. Hence the sum consists of only one term. As a result, we may understand the case k = 0 as where B 0 ({y}) = 1. This interpretation works well in the multi-dimensional cases. For example, let α, β, γ ∈ C and k 1 , k 2 , k 3 ∈ N 0 , and consider the sum (m + α) k 1 (n + β) k 2 (m + n + γ) k 3 .
This is convergent if k 1 , k 2 , k 3 ≥ 1. If some of k i 's are 0, then we modify the series. In the case when only k 1 = 0, we replace the condition m + α = 0 by m + α = 0 in the sum with the minus sign and 0 0 = 1, that is, (n + β) k 2 (m + n + γ) k 3 .
In the other cases, the sum is similarly modified. Then the special values S((k 1 , k 2 , k 3 ), (y 1 , y 2 )) for all k 1 , k 2 , k 3 ∈ N 0 are explicitly given by coefficients of a generating function, which will be given in Example 3.1.
In the above arguments the sums are taken over all integers. However in some special cases, it is possible to treat sums running over only positive integers (Examples 3.2, 3.3), which may be regarded as zeta-functions associated with hyperplane arrangements.
In the next section we will introduce more general lattice sums, and their generating functions.
For H ⊂ Λ such that rank H = r − 1, let H H = g∈H R g be the hyperplane passing through H ∪ { 0}.
The following is the main object in the present paper, a lattice sum over the hyperplane arrangement given by linear functionals belonging to Λ.
This S(k, y; Λ) is a generalization of the notion of "Weyl group symmetric" linear combinations of zetafunctions of root systems S(s, y; ∆) mentioned in the Introduction (in the case s = k); cf. [6, (3.3)], [7, (110)]. The first main result in the present paper is as follows.
In order to define the generating function of S(k, y; Λ), we need some more notations. Let B = B(Λ) be the set of all subsets B = {f 1 , . . . , f r } ⊂ Λ such that B forms a basis of V . For B ∈ B, let B * = { f B 1 , . . . , f B r } be the dual basis of B = { f 1 , . . . , f r } in V . It should be noted that for each B ∈ B, we have because all elements of Λ are indispensable for constructing a basis. Next we define a multi-dimensional generalization of fractional part {·} for real numbers, which was first introduced in [6, Section 4]. Let R = R(Λ) be the set of all subsets R = {g 1 , . . . , g r−1 } ⊂ Λ such that R = { g 1 , . . . , g r−1 } is linearly independent set. We need to fix a vector For y ∈ V , B ∈ B and f ∈ B, we define the multi-dimensional fractional part by It should be noted that for a ∈ R \ Z. Now we define the generating function of S(k, y; Λ) and state its properties.
Theorem 2.4. (i) The function F (t, y; Λ) has one-sided continuity in y ∈ V in the direction φ, that is In particular if Λ is empty, then F (t, y; Λ) is continuous on the whole V and is independent of the choice of φ.
(iii) F (t, y; Λ) is holomorphic in the neighborhood of the origin in t.
Write the Taylor expansion of F (t, y; Λ) around the origin in t as Theorem 2.5. We have The above results are again generalizations of the results proved in [6], [7]. In fact, the form of F (t, y; Λ) in Definition 2.3 is the generalization of [6, Theorem 4.1], Theorem 2.4 is the generalization of the facts mentioned in [7, p.252], and Theorem 2.5 is the generalization of [6, (3.10)].
Before going into the proofs of the main theorems, in the next section we will give several examples. Then we will start the proofs of main theorems from Section 4. Sections 4 is devoted to the proof of Theorem 2.2. Then from Section 5 to Section 8 we will describe the proof of Theorem 2.4 and Theorem 2.5. In the final section we will mention that there is some hierarchy among generating functions.

Examples
In this section we apply our theorems to some special cases, and to state explicit expressions of F (t, y; Λ), C(k, y; Λ) and S(k, y; Λ) for those examples.
Remark 3.4. We give another interpretation of the series (3.14) and (3.17), that is, we regard each term of these series as a product of positive roots of affine root system A 1 and A 2 respectively (for the theory of affine root systems, see [3]). Since there are infinitely many positive roots in affine root systems, the product consists of infinitely many factors. In order for the infinite product to make sense, we understand that infinitely many variables are set to be zero and hence the product is truncated.

Proof of Theorem 2.2
Now we start the proofs of the main theorems. First of all, in this section, we prove Theorem 2.2. The main body of the argument is the proof of an evaluation formula (Proposition 4.1) for S(k, y; Λ).
This is a generalization of [7, Theorem 6] (for integral values of k). Only the case in the domain of absolute convergence was considered in [7, Theorem 6], so there was no problem of convergence. In our present situation, if k f ≥ 2 for all f ∈ B with some fixed B ∈ B, then the matter of convergence is again obvious, so it is easy to prove our claims. However if k f = 1 for sufficiently many f ∈ Λ, then there are subtle problems on convergence, and the proof becomes much more complicated. It should be remarked that the key of the convergence of S(k, y; Λ) is the condition rank Λ = r.
Since it is difficult to treat (2.1) directly, in the following we consider a little modified sum At the last stage of the proof we will show that S(k, y; Λ) = S 1 (k, y; Λ; B 0 ). In particular, we will find that S 1 (k, y; Λ; B 0 ) actually does not depend on the choice of B 0 .
The proof of Proposition 4.1 consists of three steps.
The first step. We first consider the simplest case of (4. (Actually the sum on the left-hand side of (4.6) consists of at most one term.) The series above converge absolutely uniformly in y and hence C(k, {y}; b) and C(0, {y}; b) are continuous in y.
Proof. Let γ X,Y be the counterclockwise rectangle contour with vertices at ±X ± 2π √ −1Y . Applying the Cauchy theorem to the integral with a sufficiently small > 0, we see that the sum of all the residues vanishes, namely, and hence (4.5).
The left-hand side of (4.6) consists of only one term e −2π and hence (4.6). Both in (4.5) and (4.6), the absolute uniform convergence of the series in y is clear.
The case k = 1 is more subtle. We first prepare the following By the inequality of weighted arithmetic and geometric means (4.14) Let and C(1, {y}; b) is continuous in y. Moreover for any µ > 0 there exists K > 0 such that for all y ∈ R \ Z and all M, N ∈ R with sufficiently large |M |, |N | and M ≥ |N | Proof. From (4.1) we can easily see that (4.20) It is easy to see that there exists K > 0 independent of y such that Applying (4.21) and Lemma 4.3 to (4.20), we have for some K > 0. Therefore, choosing N = −L and M = L in (4.19) and taking the limit L → ∞, we obtain (4.16). Moreover, since C(1, {y}; b) is bounded in y, we obtain (4.17).
The second step. Secondly we consider the higher rank case of (4.3) under the special condition Λ = B with B = {B} in Lemmas 4.7, 4.8 and 4.10. We first prepare the following algebraic lemma. This statement is included in [2, Chapitre 6, Section 1, 9], but here we supply a proof.
Lemma 4.5. Let Q, P be free Z-modules of rank r with Q ⊂ P so that P/Q is a finite abelian group. Then and forλ ∈ P/Q, we have where the right-hand side denotes Kronecker's delta.
Proof. First we note that an element of Hom(P, Z) can be naturally regarded as an element of Hom(Q, Z). Denote this injection by ι. Next, let f ∈ Hom(Q, Z). It is well-known that there exist a basis {λ i } r i=1 of P and a basis {λ i } r i=1 of Q such that λ i = k i λ i with k i ∈ N and hence (4.25) where each λ i is a cyclic group of order k i withλ i ∈ P/Q. Define ϕ(f ) by the linear extension of where p denotes the natural projection Q → Q/Z. Then ϕ(f ) ∈ Hom(P/Q, Q/Z) is well-defined. We show that the sequence This implies f ∈ Hom(P, Z), and hence the exactness at Hom(Q, Z) is proved. The assertion (4.23) immediately follows from (4.27), and (4.24) follows from the orthogonality relations of group characters (cf. Apostol [1, Theorem 6.13]).
Proof. We denote the left-hand side by P and the right-hand side by Q respectively. If y ∈ P , then y = y 0 + w 0 with y 0 , f B = 0 and w 0 ∈ Z r . By setting w = −w 0 we have We prove In fact, using Lemma 4.5 with Q = Z r , P = B * and noting Hom(Z r , Z) Z r , Hom( B * , Z) Using (4.34) we find that (4.32) is equal to By Lemma 4.6 we see that the assumption y ∈ V \ f ∈Λ 1 (H Λ\{f } + Z r ) implies that y + w, f B / ∈ Z for all f ∈ Λ 1 and w ∈ Z r . Therefore the condition of Lemma 4.4 is satisfied (for y = y + w, f B ). Therefore letting N → ∞ on the right-hand side of (4.35) and applying Lemmas 4.2 and 4.4, we obtain Lastly we note that the factor { y + w, f B } on the right-hand side of the above can be replaced by .6)). This completes the proof of the lemma. Lemma 4.7 gives the right-hand side of (4.2) in the special case Λ = B and B = {B}, but for S 1 (k, y; Λ; B) instead of S(k, y; Λ). In order to use Lemma 4.7 in the proof of the general case, we need the following inequality.
and all sufficiently large N > 0, Proof. From the proof of Lemma 4.7, we have On the right-hand side of (4.38), each sum corresponding to f ∈ Λ 0 consists of just one term, and each sum corresponding to k f ≥ 2 is convergent absolutely as N → ∞, so all of them are bounded. For f ∈ Λ 1 , we apply Lemma 4.4 to obtain that the right-hand side of (4.38) is To evaluate the difference between Z(N ; k, y; Λ; B) and Z 1 (N ; k, y; Λ; B) in the final step of the proof, the following two lemmas are necessary. For B ∈ B and R > 0, let for all sufficiently large R > 0.
Proof. We show the first inclusion. Each vertex x of the parallelotope U R (B) satisfies one of the following equations where we regard (Re for all sufficiently large R > 0, where A denotes the matrix norm of A. Thus choosing We show the second inclusion. Denote by K(R) the ball of radius R whose center is the origin. The distance between the origin and the hyperplane {x ∈ V | f, x + Re holds for all sufficiently large R > 0, we find that K(R min(2 f ) −1 ) ⊂ U R (B). Therefore, choosing Proof. For brevity, we put We rearrange {f 1 , . . . , f l } = Λ + and {f l+1 , . . . , f r } = Λ 0 and decompose f ∈B with f regarded as column vectors. We rewrite the series in terms of u = Av. Let In Y j (N ) for a fixed (u 1 , . . . , u j−1 , u j+1 , . . . , u r ) with sufficiently large N ∈ N, we see that u j runs over all integers such that dN − Re Further for each j and w, we have where in the last member, we added the extra conditions |u i + Re • f i | ≤ cN for j + 1 ≤ i ≤ l, which comes from Lemma 4.9. If k j = 1, then by Lemma 4.4, and if k j ≥ 2, then directly we obtain for some K > 0. Hence for some K , K > 0. Substituting (4.57) into (4.54), we complete the proof.
The third step. Lastly we consider the general case. First we prove several preparatory lemmas.
Proof. The first assertion directly follows from the definition. Assume that f ∈ Λ. Then g, f B 0 = 0 for all g ∈ L 0 and we have which is a constant function in x. By Lemma 4.6, we find y, f B 0 / ∈ Z. This implies the second assertion. x g g, f B 0 = n for n ∈ Z, which is equivalent to Assume that f / ∈ Λ. Then there exists g ∈ L 0 such that g, f B 0 = 0. We see that (4.62) has a solution and so the equation (4.62) is rewritten as Assume that f ∈ Λ. Then by Lemma 4.11, and hence H(f, y) = ∅.
is zero.
Proof. By Lemma 4.6, we have M (y) = f ∈D w∈Z r H(f, y + w). Further by (2.3) we have so from the assumption y ∈ V \ f ∈ Λ∩D (H Λ\{f } + Z r ) we see that y + w / ∈ f ∈ Λ∩D (H B 0 \{f } + Z r ) for any w ∈ Z r . Therefore we can apply Lemma 4.12 to find that for each f ∈ D and w ∈ Z r the measure of H(f, y + w) is zero.
Lemma 4.14. Let n ∈ N and P, Q ∈ N 0 with P ≥ Q. Let a ki ∈ R for 1 ≤ k ≤ P and 0 ≤ i ≤ n such that for each k = 1, . . . , P there exists i ≥ 1 such that a ki = 0. If µ ≥ P , then a ki x i .
Since [0, 1] n is compact, by considering a neighborhood of each point x 0 in [0, 1] n and shifting the point x 0 to the origin, we see that it is sufficient to show that is finite for a sufficiently small > 0, where 0 ≤ q ≤ Q and q ≤ p ≤ P . This is estimated as For this integral, we decompose the region We show that the integral on each U k is finite. Since on U k for any m = k, we have Fix i such that a ki = 0. Then by changing variables as y i = L k (x) and y j = x j for j = i, we obtain for some r > 0. By the assumption µ ≥ P , the right-hand side is finite because Proof. By definition (4.1), for 0 ≤ x < 1, we have for sufficiently small > 0. By integrating the both sides in the region 0 ≤ x < 1, we obtain we obtain the assertion.
Proof of Proposition 4.1. Applying Lemma 4.15 with m = g, v and b = • g (for g ∈ L 0 ) to (4.4), for N > 0 we have We want to take the limit N → ∞. First we claim that it is possible to exchange the limit and the integrals. By Lemma 4.8 with µ = B 0 = r, we have say. When f ∈ Λ, then under the condition y / ∈ f ∈ Λ∩Λ 1 ∩B 0 (H B 0 \{f } + Z r ), we see that X f is just a constant because of the second assertion of Lemma 4.11. When f / ∈ Λ, by the first assertion of Lemma 4.11 we see that X f fulfills the assumption of Lemma 4.14, and hence by the lemma it is integrable since r ≥ (Λ 1 ∩ B 0 ). Therefore our claim follows form Lebesgue's dominated convergence theorem. Note that Λ ∩ Λ 1 ∩ B 0 = Λ ∩ Λ 1 because of (2.3).
Therefore from (4.83) we now obtain   The right-hand side of this equation coincides with that of (4.2). Therefore, to complete the proof of the proposition, the only remaining task is to show that S 1 (k, y; Λ; B 0 ) = S(k, y; Λ).
The sum Z(N ; k, y; Λ) has the expression which is almost the same as (4.83), only the condition v ∈ Z r ∩ U N (B 0 ) is replaced by v ∈ Z r ∩ W N . Therefore, by using Lemmas 4.9 and 4.10, we see that the difference Z(N ; k, y; Λ) − Z 1 (dN ; k, y; Λ; B 0 ) is evaluated as for some K, K > 0. Again by Lemmas 4.11 and 4.14, we obtain We have shown the convergence of S(k, y; Λ) in Proposition 4.1. Therefore to complete the proof of Theorem 2.2, we have only to show the continuity of S(k, y; Λ) in y on V \ f ∈ Λ∩Λ 1 (H Λ\{f } + Z r ).
Let y 0 ∈ V \ f ∈ Λ∩Λ 1 (H Λ\{f } + Z r ), and let G(y, (x g )) be the integrand of (4.2). Since G(y, (x g )) is bounded, we have (4.90) lim Thus it is sufficient to show that for all f ∈ Λ 1 ∩ B 0 , then (4.91) holds. Therefore it is sufficient to show that (x g ) satisfies (4.92) almost everywhere. Since Lemma 5.1. The set H R is a locally finite collection of hyperplanes, that is, for any y ∈ V there exists a neighborhood U of y such that U intersects only finitely many hyperplanes.
Proof. Let n R be a normal vector of H R . We may assume that n R ∈ Z r , because g 1 , . . . , g r−1 ∈ Z r . Then the hyperplane with v ∈ Z r can be rewritten as where m = v, n R ∈ Z and e R = n R / n R , n R . Therefore and so Hence the assertion follows from this expression and R < ∞.  Proof. By Lemma 5.1, for any y ∈ V , we see that y + cφ ∈ H R and so y + cφ, f B ∈ Z for all sufficiently small c > 0. Therefore, if y, f B ∈ Z, then by (2.6). If y, f B ∈ Z, then Hence we have the assertion.
By this lemma we immediately obtain This shows the assertion (i) of Theorem 2.4. Next, observe that the right-hand side of (4.2) can be defined for any y ∈ V (though (4.2) itself is valid only under the assumption of Proposition 4.1). Therefore, we can define C(k, y; Λ) for any y ∈ V as the multiple of the right-hand side of (4.2), and we introduce the generating function of C(k, y; Λ) of the form A more explicit form of the generating function can be deduced by substituting the formula of Proposition 4.1 into (5.10). In fact, Lemma 5.3. For any y ∈ V , the series on the right-hand side of (5.10) is absolutely and uniformly convergent in the neighborhood of the origin with respect to t ∈ C Λ . Furthermore we have Proof. The following proof is similar to that of [7,Lemma 7]. By (4.80) we see that, for b ∈ C, there exists a sufficiently small R b > 0 such that holds for y ∈ R. Thus we have for 0 ≤ y ≤ 1 where (5.14) Fix r such that 0 < r < min f ∈Λ R • f . Then for |t f | < r (f ∈ Λ), we have the uniform and absolute convergence of F (t, y; Λ), which implies the holomorphy of F (t, y; Λ) in the neighborhood of the origin with respect to t ∈ C Λ . Furthermore by exchanging the sum and the integral and using (4.1) we obtain which yields (5.11).
Lemma 5.4. F (t, y; Λ) is continuous in y on V \ f ∈ Λ (H Λ\{f } + Z r ) and has one-sided continuity in y ∈ V in the direction φ.
Proof. The proof is almost the same as that of the continuity of S(k, y; Λ) in (4.90). Let G(y, (x g )) be the integrand of the last expression of (5.11). In this case, the continuity comes from (4.92) for all f ∈ B 0 . Hence the first assertion follows from Lemma 4.13 with D = B 0 . The second assertion immediately follows from Lemma 5.2.
We have obtained the assertions, corresponding to (i), (ii) and (iii) of Theorem 2.4, for F (t, y; Λ). In the following sections, we will prove for y ∈ V \ H R . Then F (t, y; Λ) = F (t, y; Λ) on the whole V by the one-sided continuity of F (t, y; Λ) and F (t, y; Λ) which we have already shown. Thus automatically the assertions (ii) and (iii) of Theorem 2.4 will follow. Also, comparing (2.9) with (5.10), we find that (5.19) C(k, y; Λ) = C(k, y; Λ).
By the definition of C(k, y; Λ) and Proposition 4.1, we have for y ∈ V \ f ∈ Λ∩Λ 1 (H Λ\{f } + Z r ). Combining this with (5.19), we obtain the assertion of Theorem 2.5. Therefore the only remaining task is to show (5.18) for y ∈ V \ H R .

The generating function and convex polytopes
The aim of the following three sections is to prove (5.18), which will be shown in Section 8. The present and the next sections are devoted to the preparations for the proof of (5.18), which are connected with the theory of convex polytopes.
First, we summarize some definitions and facts about convex polytopes (see [6,7,13]). For a subset X ⊂ R N , we denote by Conv(X) the convex hull of X. A subset P ⊂ R N is called a convex polytope if P = Conv(X) for some finite subset X ⊂ R N . Let P be a d-dimensional polytope. Let H be a hyperplane in R N . Then H divides R N into two half-spaces. If P is entirely contained in one of the two closed half-spaces and P ∩ H = ∅, then H is called a supporting hyperplane of P. For a supporting hyperplane H and a subset F = P ∩ H = ∅, the subset F is called a face of the polytope P and H a supporting hyperplane associated with F. If the dimension of a face F is j, then we call it a j-face F. A 0-face is called a vertex, a 1-face an edge and a (d − 1)-face a facet. For convenience, we regard P itself as its unique d-face. Let Vert(P) be the set of all vertices of P. Then for a face F. A d-dimensional simple polytope is a polytope whose vertices are adjacent to exactly d edges.
The definition of polytopes above is that of "V-polytopes". We mainly deal with another representation of polytopes, "H-polytopes" instead, that is, a bounded subset of the form It is known (Weyl-Minkowski) that H-polytopes are V-polytopes and vice versa.
We have an expression of k-faces in terms of hyperplanes H i = {x ∈ R n | a i · x = h i }. Then we have . Now we present a fundamental proposition, which gives an expression of F (t, y; Λ) involving integrals over certain convex polytopes. This proposition is a generalization of [7, Theorem 7]. Proposition 6.4.
is a convex polytope or an empty set.
Proof. We fix a representative of each w ∈ Z r / B 0 in Z r . Let m = (m f ) f ∈B 0 ∈ Z r , and denote by Q(w, m) the set of all x = (x g ) g∈L 0 satisfying the conditions 0 ≤ x g ≤ 1 (g ∈ L 0 ) and This condition is equivalent to Denote the multiple integral on the right-hand side of (5.11) by I(w), and divide it as . Applying (6.10), we obtain (6.11) Note that w + h∈B 0 m h h runs over Z r , when w ∈ Z r / B 0 and m ∈ Z r run. Therefore, rewriting w + h∈B 0 m h h as m, we obtain the assertion of the proposition.
Remark 6.5. For readers' convenience, we give typical pictures of P(m; y) in the cases y ∈ H R and y / ∈ H R , which will be treated below in Lemmas 7.1, 7. L 0 = {(ae 1 + be 2 , α g ), (ce 1 + de 2 , α h )}, (6.14) where a, b, c, d are positive integers, which corresponds to the series (6.16) S(k, y; Λ) = n 1 ,n 2 e 2π √ −1(n 1 y 1 +n 2 y 2 ) (n 1 + α 1 ) k 1 (n 2 + α 2 ) k 2 (an 1 + bn 2 + α g ) kg (cn 1 where α 1 , α 2 , α g , α h ∈ C, k 1 , k 2 , k g , k h ≥ 2 and n 1 , n 2 run over all integers such that the denominator does not vanishes. Then we have In the case a = b = c = 1, d = 2, the polytope P(m; y) is drawn as in Figure 2 if y ∈ H R and in Figure 3 if y / ∈ H R . In the former case, there are more than 2 (= Λ − r) hyperplanes at some vertices while in the latter case, there are only 2 hyperplanes at each vertex, which ensures that P(m; y) is a simple polytope in higher dimensions. The argument developed in this and the next sections is a generalization of that in [6,Section 6]. Let A = {0, 1} Λ−r . Let B be the set of all subsets of Λ which have r elements, and define W = B × A and W = B × A . Obviously W ⊂ W . For an element W = (B, A) ∈ W , we number A = (a g ) g∈Λ\B . We fix a decomposition Λ = B 0 ∪ L 0 with B 0 ∈ B, and for f ∈ Λ, a ∈ {0, 1}, m ∈ Z r and y ∈ V , we define u(f, a) ∈ R L 0 by where g runs over L 0 , and define v(f, a; m; y) ∈ R by Further we define the hyperplanes where for w = (w g ), x = (x g ) ∈ C L 0 , we have set Then we have H + (f, a; m; y). Lemma 7.1. All vertices of P(m; y) are of the form Proof. By Proposition 6.2, we see that any vertex of P(m; y) is obtained as the intersection of ( L 0 ) hyperplanes. Since for f ∈ Λ, two hyperplanes H(f, a; m; y) (a = 0, 1) are parallel and hence their intersection is empty, we see that a vertex must be of the form (7.7).
Since L = Λ − r, we have B = Λ \ L ∈ B . Therefore Lemma 7.1 implies that any vertex of P(m; y) determines an element (B, A) ∈ W . The next lemma is a kind of converse assertion. Proof. Let B = {f 1 , . . . , f r } ∈ B and a f ∈ {0, 1} for f ∈ L = Λ \ B. Consider the intersection of ( Λ − r) hyperplanes (7.8). Then this set consists of the solutions of the system of the ( L 0 ) linear equations (7.9) g∈L 0 x g g, ( where E p is the p × p identity matrix. H + (f, a; m; y).
Proof. By (7.6), we see that P(m; y) is defined by ( Λ) pairs of inequalities. By Proposition 6.1, the point p(m; y; W ) is a vertex of P(m; y) if and only if all of these inequalities hold. We see that ( L 0 ) pairs among them are automatically satisfied, because . Divide the left-hand side of (the first formula of) (7.9) and (7.17) into two parts according to the conditions g ∈ B \ B 0 and g ∈ L 0 \ B (with noting L 0 = (B \ B 0 ) ∪ (L 0 \ B)). Then we obtain an overdetermined system with the p variables x g for g ∈ B \ B 0 and the (p + 1) equations (7.18) g∈B\B 0 x g g, Hence we have where (x g ) g∈B\B 0 is a row vector and M (y) is a (p + 1) × (p + 1) matrix defined by As the consistency for these equations, we get det M (y) = 0. We may rewrite and g, f B 0 = 0 for all g on the right-hand side.
Since the row vectors ( g, f B 0 ) f ∈(B 0 \B)∪{h} for g ∈ B \ B 0 are linearly independent, det M (y) = 0 implies that the last row vector is written as a linear combination of the other row vectors. That is, for f ∈ (B 0 \ B) ∪ {h} we have (7.23) y + m − g∈(Λ\B)∪{h} with some q g ∈ R, and so Vectors which are orthogonal to all f B 0 (f ∈ (B 0 \B)∪{h}) are spanned by g (g ∈ (B ∩B 0 )\{h}). Therefore, since m − g∈(Λ\B)∪{h} a g g ∈ Z r , we have which implies the desired claim.
Next we consider the condition p(m; y; W ) ∈ H(h, a h ; m; y) for some h ∈ B \ B 0 and a h ∈ {0, 1}. Then similarly as above, we see that y lies in H R because This completes the proof of the lemma. Lemma 7.6. Let y ∈ V \ H R and W ∈ W . Then we have where t * = (t * g ) g∈L 0 with t * g defined by (6.6).
Proof. By (7.8), the point p(m; y; W ) = (x g ) g∈L 0 satisfies By Lemma 7.2, the system of these equations has a unique solution.
In the case f ∈ B 0 \ B, we have On the other hand, in the case f ∈ B 0 ∩ B we have In fact, since holds for any z ∈ V , we have Here we note that for h ∈ B \ B 0 , holds. Because, using (7.32), for f ∈ B 0 \ B we obtain (7.35) Comparing (7.30) with (7.35), we obtain (7.34) due to the uniqueness of the solution.
By the first equality of (7.30), the last term on the last member of (7.36) is equal to On the other hand, for f ∈ B 0 ∩ B, we have by (7.34) and (7.31). Therefore we finally obtain (7.28).
Lemma 7.7. Let y ∈ V \ H R and W ∈ W . Then the point p(m; y; W ) is a vertex of P(m; y) if and only if Proof. By Lemma 7.3, the point p(m; y; W ) = (x g ) g∈L 0 is indeed a vertex if and only if For f ∈ B ∩ B 0 , applying (7.29) (the second equality) and (7.34), we have g∈L 0 (7.41) Therefore, noting (7.31), we see that the first pair of inequalities of (7.40) is which is equivalent to For f ∈ B \ B 0 , noting (7.34) we see that the second pair of inequalities of (7.40) is again of the same form as (7.42). Therefore the desired assertion follows.
Fix W = (B, A) ∈ W . Let U be the ( L 0 ) × ( L 0 ) matrix whose f -th column consists of u(f, a f ) for f ∈ Λ \ B and U (h, v) be the matrix U with only the h-th column replaced by v. Then we have the following two lemmas.
Proof. By rearranging rows and columns we see that Proof. We show that x = (x f ) f ∈Λ\B defined by (7.47) is a unique solution of the linear equation Then the statement follows from Cramer's rule. 8. Completion of the proof of Theorem 2.4 and Theorem 2.5 In this section we prove (5.18) to complete the proof of Theorems 2.4 and 2.5. First we show an elementary lemma.
Let n ∈ N and let {e 1 , . . . , e n } be a standard basis of R n . Let M = (m ij ) 1≤i≤n 1≤j≤n be an n × n matrix. For 1 ≤ j ≤ n, and v ∈ R n , let M (j, v) be the matrix M with only the j-th column replaced by v. Let where b ij is the cofactor of (i, j)-entry of M . The following lemma (properties of cofactor matrices) is shown by elementary linear algebra, and we omit the proof.
(1) For v ∈ R n , we have (2) Let U = (u(j)) 1≤j≤n be the n × n matrix whose j-th column consists of u(j). Then Now we start the proof of (5.18), that is F (t, y; Λ) = F (t, y; Λ) for y ∈ V \ H R . Since all the polytopes P(m; y) for y ∈ V \ H R are empty or simple by Lemma 7.4, we may apply Lemma 6.3 to the right-hand side of Proposition 6.4. In the present case the vertices are of the form p(m; y; W ), satisfying (7.39) by Lemma 7.7. Therefore we have This h is unique because otherwise we have p(m; y; W ) = p(m; y; W ). Since E(m; W ) = (Λ \ B) = Λ − r (because P(m; y) is simple), we find that there is a one-to-one corresponding between E(m; W ) and Λ \ B.
By (8.7), we see that the set of the equations and the inequality with respect to v (8.9) u(f, a f ) · v = 0 (f ∈ Λ \ (B ∪ {h})), u(h, a h ) · v < 0 has a solution v = p(m; y; W ) − p(m; y; W ).
We construct another vector e(m; W, W ) satisfying (8.9) so that .
The construction of e(m; W, W ) is as follows. Let e g for g ∈ L 0 be the standard orthonormal basis of R L 0 . Let U be the ( Λ − r) × ( Λ − r) matrix whose f -th column consists of u(f, a f ) for f ∈ Λ \ B with W = (B, A). For h ∈ Λ \ B, let U (h, v) be the matrix U with only the h-th column replaced by v. Note that det U = 0 by Lemma 7.8. Define We observed that h runs over Λ \ B when W runs over E(m; W ). Therefore by Lemma 8.1(2) we see that (8.12) implies | det(e(m; W, W )) W ∈E(m;W ) | = | det U | Λ−r−1 .