CHARACTERIZING f-VECTOR

AHMED UMER ASHRAF

Abstract. Given a d-dimensional convex polytope, its kth face number is the number of (k − 1)-dimensional faces it has. The f-vector of a polytope is the sequence of its face numbers. Beside Euler’s formula, these numbers satisfy further equalities and inequalities. Characterization of f-vector of d-dimensional convex polytope is already known for d ≤ 3. For d ≥ 3, we do not have a complete answer, but g-theorem gives us a characterization for simplicial (and dually simple) case. Here we review g-theorem and its various proofs.

Contents 1. Prehistory 1 2. Klee-Dehn-Sommerville Theorem4 3. Kruskal-Katona-Sch¨utzenberger Theorem5 4. Stanley-Macaulay Theorem6 5. Necessity of g-theorem7 5.1. Stanley’s approach7 5.2. McMullen’s approach8 5.3. Timorin’s presentation9 6. Frontiers 10 References 11

1. Prehistory There has been an interest in convex polytopes from the time of Greeks as far as we know. This is evident from complete characterization of Platonic solids in Euclid’s Elements, which are examples of 3-dimensional polytopes. Euler in 1752 observed the remarkable relation between number of vertices v, edges e and faces f of a convex 3-dimensional polytope v − e + f = 2 Though there is evidence that Descartes already knew this around 1640. This formula was generalized by Schl¨afli[Sch50] for arbitrary d-dimensional convex polytopes in 1852. Schl¨afli in his proof assumed shellability of convex d-polytopes which was not proved until 1971 by Bruggesser and Mani [MB71]. Though before that Poincar´e[Poi93] developed the machinery of Algebraic topology and gave the first proof in 1893 of what now is known as Euler-Poincar´e- Schl¨afliformula d X k+1 (−1) fk = 0 k=0 for any convex d-polytope P , where fk is number of k − 1 dimensional faces of P . 1 2 AHMED UMER ASHRAF

Definition 1.1. Given an abstract simplicial complex or a convex polytope K, we can define the f-vector of K as

f(K) = (f0, f1, f2,... ) where  fk = # σ ∈ K : dim(σ) = k − 1 i.e. the number of (k − 1)-dimensional faces. We call fk’s the face numbers of K. We also define the f-polynomial of K to be

dim(K)+1 X dim(σ)+1 X k fK(t) = t = fkt σ∈K k=0 Example 1.2. Following are some face numbers computed via elementary counting arguments

• For 3-simplex ∆3, we have f(∆3) = (1, 4, 6, 4, 1) and in general  d + 1 d + 1 d + 1 d + 1  f(∆ ) = 1, , ,..., , , 1 d 1 2 d − 1 d d+1 f∆d (t) = (1 + t)

• For 3-cube 3, we have f(3) = (1, 8, 12, 6, 1), and in general  d d  d   f( ) = 1, 2d , 2d−1 ,..., 2 , 1 d 0 1 d − 1 f (t) = 1 + t(2 + t)d d

• For octahedron ♦3, we have f(♦3) = (1, 6, 12, 8, 1) and in general  d d d  f( ) = 1, 2 , 22 ,..., 2d , 1 ♦d 1 2 0 f (t) = (2t + 1)d + td+1 ♦d

Notice that the duality between the f(d) and f(♦d) is the consequence of the duality between d and ♦d in general. ♦ d+2 Let’s ask ourselves what vector (f0, f1, ··· , fd, fd+1) ∈ Z can arise as f-vectos of a d- dimensional convex polytope. Note that f0, fd+1 are always 1. For the sake of brevity, let’s denote v := f1, e := f2 and f := f3, for the immediate discussion • For d = 0, the only convex polytopes are points, which means the only f-vector is (1, 1). • For d = 1, the only convex polytopes are closed intervals, which implies the only f-vector is (1, 2, 1). • For d = 2, the only convex polytopes are convex polygons, whose f-vector can be characterized by vectors of the form (1, v, e, 1) such that v = e v, e ≥ 3 CHARACTERIZING f-VECTOR 3

• For d = 3, we have a characterization of f-vector of 3-polytopes due to Ernst Steinitz [Ste10] from 1922. It says that any vector of the form (1, v, e, f, 1), satisfying v − e + f = 2 (Euler’s formula) v, f ≥ 4 2e ≥ 3f, 2e ≥ 3v is an f-vector of a 3-polytope and vice versa. • For d = 4, we don’t have a complete answer yet. Notice that the Euler relation is a constrained from topology and the inequalities are the constrained from convexity of polytopes. Definition 1.3. The h-vector of K as

h(K) = (h0, h1, h2,... ) where k X d − i h = (−1)k−if k i k − i i=0 where fi’s are face numbers of K. More compactly, we can define    i+j d − j Md = (−1) i − j 0≤i,j≤d then the definition read as

h(K) = Mdf(K) We also define the h-polynomial of K to be

dim(K)+1 X k hK(t) = hkt k=0 Example 1.4. Going back to our running example of simplex, cube and octahedron, we have

• For 3-simplex ∆3, we have h(∆3) = (1, 0, 0, 0, 0) and in general

h(∆d) = (1, 0,..., 0)

h∆d (t) = 1

Notice that h(∂∆d) = (1, 1,..., 1). • For 3-cube 3, we have h(3) = (1, 4, −6, 2, 0), and

h(∂3) = (1, 5, −1, 1)

• For octahedron ♦3, we have f(♦3) = (1, 2, 0, −2, 0) and

h(∂♦3) = (1, 3, 3, 1) ♦ Remark 1.5. Keeping in view the definition of f-vector and h-vector of a simplicial complex, we should notice the following facts: 4 AHMED UMER ASHRAF

• f0 = h0 = 1 for all simplicial complexes. Further, one may notice

f∆(−1) = −χe(∆) d hd = (−1) χe(∆)

• The transformation matrix Md is invertible with inverse given by   −1 d − j Md = i − j 0≤i,j≤d • Another way to state the relationship between f-polynomial and h-polynomial is d X  t  f(t) = h ti(1 + t)d−i = (1 + t)d · h i 1 + t i=0 d X  t  h(t) = f ti(1 − t)d−i = (1 − t)d · f i 1 − t i=0

2. Klee-Dehn-Sommerville Theorem Noticing the palindromicity of h(K), whenever K happen to be the boundary of a simplical polytope, Max Dehn [Deh04] (in 1904) and Duncan Sommerville [Som27] (in 1927), showed that hk(K) = hd−k(K) for all k = 0, . . . , d, whenever K is a simplicial sphere of dimension (d − 1). Victor Klee [Kle64a] generalized this result in 1964, to any Eulerian complex of dimension (d − 1). Here, we state the formulation of Klee. Theorem 2.1. (Klee-Dehn-Sommerville Equations, KDS equations) [Kle64a, Deh04, Som27, Sta12] Suppose ∆ is a pure Eulerian complex of dimension d − 1 and h(∆) = (h0, h1, . . . , hd), where h0 = 1, then for each k ≥ 0 d  h − h = (−1)k 1 − (−1)dχ(∆) k d−k k e d−1 In paricular, when χe(∆) = (−1) , then

hk = hd−k We also mention the generalization of KDS for Eulerian posets. Since face lattice of convex polytopes is also Eulerian, this can be viewed as KDS for convex polytopes. For its formulation, we need to define the graded version of f-vector Definition 2.2. Let P be a finite graded poset of rank n, with rank function rk : P −→ [n].

• The flag f-vector (fS : S ⊆ [n]) is defined as

fS = number of chains C such that rk(C) is given by S

• The flag h-vector (hS : S ⊆ [n]) is defined as X |S|−|T | hS = (−1) fT T ⊆S • The ab-index of P is defined as X ΦP (a, b) = hSuS S⊆[n] CHARACTERIZING f-VECTOR 5

where uS = u1u2 ··· un and ( a if i∈ / S ui = b if i ∈ S Theorem 2.3. [BB85, BK91, Sta94] The ab-index of an Eulerian poset P , can be expressed as a unique polynomial Ψ(c, d) called the cd-index of P , where c = a + b d = ab + ba Furthermore, if P is the face poset of a polytope, then the coefficients of Ψ(c, d) are positive. So now we can state the analogue of KDS equations in case of general convex polytope. Corollary 2.4. For an Eulerian poset, we have

hS = hSc where Sc denote the complement of S in [n].

3. Kruskal-Katona-Schutzenberger¨ Theorem There is a complete characterization of f-vector of abstract simplicial complexes given by Kruskal-Katona-Sch¨utzenberger Theorem. These are worked out by Joseph Kruskal in 1963 [Kru63] and Gyula Katona in 1966 [Kat68]. Though Marcel-Pail Sch¨utzenberger already described them in 1959 in a technical report for MIT’s Research Laboratory of Electronics [Sch59]. To state these, we need a couple of definitions:

Theorem 3.1. Given n, k ∈ Z+, there exists a unique representation of n in the form a  a  a  n = k + k−1 + ··· + r k k − 1 r where ak > ak−1 > . . . > ar ≥ r ≥ 1. This representation of n is call the k-cascade representation of n. Given such a representation of n, define

• kth Kruskal-Katona function ∂k

∂k : N −→ N 0 7→ 0 a  a  a   a  a   a  n = k + k−1 + ··· + r 7→ k + k−1 + ··· + r k k − 1 r k − 1 k − 2 r − 1 • kth Macaulay function ∂k k ∂ : N −→ N 0 7→ 0 a  a  a  a − 1 a − 1 a − 1 n = k + k−1 + ··· + r 7→ k + k−1 + ··· + r k k − 1 r k − 1 k − 2 r − 1 • We call n ∈ N(∞) a K-sequence if

∂k(nk) ≤ nk−1 for all k ≥ 1. 6 AHMED UMER ASHRAF

• We call n ∈ N∞ an M-sequence if k ∂ (nk) ≤ nk−1 for all k ≥ 1. Theorem 3.2. (Kruskal-Katona-Sch¨utzenberger theorem, KKS) [Kru63, Kat68, Sch59] f ∈ N∞ is a K-sequence if and only if it is the f-vector of a simplicial complex. 4. Stanley-Macaulay Theorem Before we mention Stanley-Macaulay’s generalization of this theorem, let us review Stanley Reisner ring of a complex. Let ∆ be an abstract simplicial complex on vertex set [n]. Consider the polynomial ring K[x1, x2, . . . , xn], we can define the Stanley-Reisner Ideal

I∆ = hxτ | τ∈ / ∆i where xτ := xi1 xi2 . . . xir for τ = {i1, i2, . . . , ir}. Since I EK[x1, x2, . . . , xn], we define Stanley- Reisner Ring associated to ∆ as

k[∆] := R/I∆

Notice that I∆ is a homogenous ideal for any simplicial complex ∆, this implies the k[∆] inherits the same grading as R. We recall the following facts: • For any simplicial complex ∆, we have dim(K[∆]) = dim(∆) + 1. • Primary decomposition of I∆ is given by

I∆ = pF1 ∩ pF2 ∩ ... pFm

where F1,F2,...,Fm are the facets of ∆. • If K[∆] is Cohen-Macaulay, then ∆ is pure. • The set of simplicial complexes ∆ on [n] is in bijection with the set of square free ideals I E K[x1, . . . , xn]. Theorem 4.1. (Stanley-Macaulay’s theorem [Sta96][Mac]) A sequence f ∈ N∞ is an M- sequence if and only if any one (and therefore all) of the following equivalent conditions hold • f is the f-vector of a multicomplex. • f is the Hilbert function of a finitely generated standard N-graded algebra over K. • f is the Hilbert function of a CM standard N-graded algebra over K. • f is the h-vector of CM simplicial complex. • f is the h-vector of a shellable simplicial complex. From this we get the Upper Bound conjecture of Motzkin [Mot57] as a corollary, proved first by McMullen [McM70]. The conjecture and the proof was given first for convex d-polytopes. Klee [Kle64b] conjecture the statement in generality of simplicial sphere, which was later proved by Stanley [Sta75] Theorem 4.2. (Upper Bound Theorem [Mot57, Kle64b, Sta75, McM70]) If ∆ is a d − 1  d  simplicial sphere with n vertices, then for all i ≤ 2

fi(∆) ≤ fi(Cn,d) where Cn,d is the cyclic polytope on n vertices in d dimension, or equivalently, n − d + i − 1 h (∆) ≤ i i CHARACTERIZING f-VECTOR 7

Proof. If ∆ is a sphere, then K[∆] is Cohen-Macaulay [Rei76] which by Stanley-Macaulay theorem implies h(∆) is an M-vector. Since ∆ has n vertices and dimension d − 1 so h1(∆) = n − d. Since h is an f-vector of a multicomplex for which we know the i-th co-ordinate is n−d+i−1
bounded by i , implying the required result.  Now we tackle the problem of P being a simplicial polytope, instead of characterizing the f-vector of polytopes in general. Because in general things seems not that easy. McMullen [McM71] conjectured the following characterization in 1971 for simplicial polytopes

∞ Theorem 4.3. A sequence h = (h0, h1, . . . , hd) ∈ N is the h-vector of a simplicial d-polytope P if and only if the following conditions are being satisfied

• Dehn-Sommerville equations: hi = hd−i. • g = (g0, g1, . . . , g d ) is an M-vector, where b 2 c

g0 = 1 d g = h − h for i = 1, 2,..., i i i−1 2 It’s sufficiency was proved by Billera and Lee [BL80][BL81] in 1980 by a clever construc- tion and it’s necessity was proved by Stanley [Sta80]. In remainig part of this paper we’ll concentrate on the later.

5. Necessity of g-theorem 5.1. Stanley’s approach. Let P be an integral simplicial convex d-polytope with vertex set n AP = {a1, a2,..., am} ⊂ Z . This defines a map ∗ n m ϕA :(C ) −→ P t 7−→ (ta1 : ta2 : ··· : tam )

The Zariski closure of the image of ϕA is called projective toric variety XP associated to P i.e.

XP = im(ϕA) ∗ 2i Let H (XP ) denote the singular cohomology ring of XP over R. Further dim(H (XP )) = hi(P ). Then Stanley used the following result to prove the necessity Theorem 5.1. Let P be an integral simplicial convex d-polytope in Rd and let ∆ = ∂P . Let XP be the corresponding toric variety • There is an algebra isomorphism ∗ κ : H (XP ) −→ R := R[∆]/(θ1, θ2, . . . , θd)

for certain lsop θ1, θ2, . . . , θd of R[∆]. 2 • (Hard Lefschetz Theorem) There exists ω ∈ H (XP ) such that d−2i 2i 2d−2i ×ω : H (XP ) −→ H (XP ) is an isomorphism. From the first part, we see that 2 ∗ Hilb(R, t ) = Hilb(H (XP ), t) 8 AHMED UMER ASHRAF

2i and so dim(H (XP )) = hi. It trivially follows h0 = 1. From Poincar´eduality, we also get the KDS equations

hi = hn−i To show that g = ∆h is an M-vector, we need to construct a finitely generated standard 2i graded algebra, which can be done using the second part. Identifying Ri with H (XP ), we can say

ω : Ri−1 −→ Ri  d  is injective for 0 ≤ i ≤ 2 , which implies Hilb(R/ωR, i) = Hilb(R, i) − Hilb(ωR, i) = Hilb(R, i) + Hilb(R, i − 1)

= hi − hi−1

= gi Hence we have a finitely generated standard N-graded algebra with g being it’s Hilbert function. 5.2. McMullen’s approach. Stanley’s proof of necessity is remarkably brilliant but there was a desire of purely combinatorial proof of this fact. For precisely that purpose McMullen came up with combinatorial version of Hard Lefschetz theorem in [McM77] and subsequently improving and correcting it in [McM93].

Definition 5.2. Let Π be the abelian group with a generator [P ] for each polytope P in Rd, along with the following relations • (Valuation): For each P,Q ∈ P(Rd), we have [P ∪ Q] + [P ∩ Q] = [P ] + [Q], whenever P ∪ Q is a polytope. • (Translation Invariance): For each P ∈ P(Rd) and t ∈ Rd, we have [P + t] = [P ]. We can define mutliplication in Π induced by Minkowski sum as [P ] · [Q] := [P + Q]

We can also define the dilatation operator dλ for each λ ∈ R as

dλ[P ] := [λP ] On this construction, McMullen proved the following structure theorem: Theorem 5.3. [McM77] (McMullen’s decomposition theorem) Π is almost a graded algebra over R, in the following sense • There is a direct sum decomposition d M Π = Πk k=0 ∼ ∼ with Π0 = Z and Πk is a real vector space for k = 1, . . . , d with Πd = R. • For a, b ≥ 0, we have

Πa · Πb = Πa+b

with Πk := {0}, for k > d. + + d • (λa)b = λ(ab) = a(λb) for all a, b ∈ Π and λ ∈ R, where Π := ⊕k=1Πk. k • dλa = λ a for a ∈ Πk and for λ ≥ 0, where k = 1, 2, . . . n. CHARACTERIZING f-VECTOR 9

Since [P ] − [o] ∈ Π+ is nilpotent for all [P ] ∈ Π, define ∞ X (−1)k p := log[P ] = ([P ] − [o])k k k=1 Let P is simple d-polytope, consider the subalgebra Π(P ) of Π generated by the class of all the Minkowski summands of P , then we have the following analogue of Theorem 5.1 d Theorem 5.4. Let P be a simple d-polytope, let p = log(P ) and let 0 ≤ k ≤ 2 , then • There’s an isomorphism d−2k p :Πk(P ) −→ Πd−k(P )

• Π(P )/ hpi is a standard algebra whose kth graded subspace has dimsnion gk(P ) for  d  each k = 0,..., 2

Using an isomorphic constructon of polytope algebra, McMullen showed that dim(Πk(P )) = hk. This then combined with the first part of the theorem gives palindromicity of h vector (KDS equations). Now to show the second part of g-theorem, we go in the same manner as in Stanley’s proof and get

dim(Π(P )/ hpii) = gi which then implies that g is an M-vector. 5.3. Timorin’s presentation. Motivated by the work of Khovanskii and Pukhilov [PK92], Timorin gave another presentation of polytope algebra in [Tim99]. Here we outline his ap- d proach to prove the g-conjecture. Given a polytope P let PP (R ) denote the set of all polytopes d in R analogous to P . Notice that PP is closed under taking Minkowski sum and multiplica- tion by positive scalar. This implies that PP has a structure of convex cone. We can extend it to make a vector space by considering formal differences of polytopes. For the rest of this section, we’ll denote this vector space by PP . Hopefully there will be no confusion because we are not using the cone again in this section.

Definition 5.5. Let P be a polytope with facets F1,F2,...,Fk and let n1, n2,..., nk be their 0 respective outward normal covectors. Let us consider some polytope P ∈ PP , then it can be characterized by it’s support numbers 0 ηi(P ) = max hni, pi p∈P These are just signed distances from the origin to the supporting hyperplane containing the facets. The functions η1, ··· , ηm form a system of co-ordinates in PP .The restriction of the volume polynomial to PP can be expressed as a polynomial

Vol = VolP (η1, η2, ··· , ηm) Definition 5.6. Let Diff denote the algebra of differential operators with constant coefficients on PP . It is a commutative polynomial ring with generators ∂ ∂i := ∂ηi for i = 1, . . . , m. Notice that it has a natural grading M Diff = Diffk k∈N 10 AHMED UMER ASHRAF where Diffk denote the homogeneous component consisting of differential operators of degree k. Let I ⊂ Diff be the annihilator of Vol in Diff i.e. I(Vol) = {∂ ∈ Diff : ∂Vol = 0} Now define the algebra corresponding to the polynomial Vol A(Vol) := Diff/I(Vol) Since Vol is a homogeneous polynomial, the ideal I is graded and so is the ring A(Vol) i.e. n M A(Vol) = Ak(Vol) k=0 Then Timorin goes on to prove the following analogue of Hard Lefschetz theorem Theorem 5.7. [Tim99] Given any convex simple polytope P , • We have a nondegenerate pairing

h·, ·i : Ak(P ) ⊗ An−k(P ) −→ R α ⊗ β 7−→ αβVol Further

dim(Ak) = hk(P ) • Define the Lefschetz operator in A(P ) m X LP = ni∂i ∈ A1(P ) i=1 n then for all i ≤ 2 , n−2i L : Ai(P ) −→ An−i(P ) is an isomorphism. From this the g-theorem follows in the same manner as in McMullen’s proof.

6. Frontiers Beside the obvious general problem of characterization of convex d-polytope, there are a lot of other open problems in this area. Here we mention some to get a flavor: • Boundary of convex simplicial d-polytopes, are simplicial spheres, whose face vector is already being characterized by g-theorem. But if we consider simplicial spheres in general, then we know that they are Eulerian complex and hence satisfy KDS equations. But we still don’t have a characterization of their f-vectors. • A finite abstract polyhedral complex can be considered as finite meet-semilattice. We call it a cubical complex if every interval [0ˆ, x] is isomorphic to the face lattice of a cube (of some dimension d(x)). We can ask analogue of KKS for cubical complexes. • Flag complex is a simplicial complex whose minimal nonfaces are just edges. Charney- k Davis conjecture says if ∆ is a (2k−1)-dimensional flag sphere then (−1) h∆(−1) ≥ 0. • A simplicial complex whose restriction to any subset of ground set is pure, is called matroid complex. We already know that matroid complex are shellable and hence CM. By Stanley-Macaulay, we know it’s h-vector is an f-vector of multicomplex. Stanley asked whether it is an f-vector of a pure multicomplex? CHARACTERIZING f-VECTOR 11

References [BB85] Margaret M. Bayer and Louis J. Billera, Generalized Dehn-Sommerville relations for polytopes, spheres and Eulerian partially ordered sets, Invent. Math. 79 (1985), no. 1, 143–157. MR 774533 (86f:52010b) 2.3 [BK91] Margaret M. Bayer and Andrew Klapper, A new index for polytopes, Discrete Comput. Geom. 6 (1991), no. 1, 33–47. MR 1073071 (91k:52024) 2.3 [BL80] Louis J. Billera and Carl W. Lee, Sufficiency of McMullen’s conditions for f-vectors of simplicial polytopes, Bull. Amer. Math. Soc. (N.S.) 2 (1980), no. 1, 181–185. MR 551759 (81b:52004)4 [BL81] , A proof of the sufficiency of McMullen’s conditions for f-vectors of simplicial convex poly- topes, J. Combin. Theory Ser. A 31 (1981), no. 3, 237–255. MR 635368 (82m:52006)4 [Deh04] M. Dehn, Zwei Anwendungen der Mengenlehre in der elementaren Geometrie, Math. Ann. 59 (1904), no. 1-2, 84–88. MR 15112612, 2.1 [Kat68] G. Katona, A theorem of finite sets, Theory of graphs (Proc. Colloq., Tihany, 1966), Academic Press, New York, 1968, pp. 187–207. MR 0290982 (45 #76)3, 3.2 [Kle64a] Victor Klee, A combinatorial analogue of Poincar´e’sduality theorem, Canad. J. Math. 16 (1964), 517–531. MR 0189039 (32 #6466)2, 2.1 [Kle64b] , On the number of vertices of a convex polytope, Canad. J. Math. 16 (1964), 701–720. MR 0166682 (29 #3955)4, 4.2 [Kru63] Joseph B. Kruskal, The number of simplices in a complex, Mathematical optimization techniques, Univ. of California Press, Berkeley, Calif., 1963, pp. 251–278. MR 0154827 (27 #4771)3, 3.2 [Mac] F. S. MacAulay, Some Properties of Enumeration in the Theory of Modular Systems, Proc. London Math. Soc. S2-26, no. 1, 531. MR 1576950 4.1 [MB71] P Mani and H Bruggesser, Shellable decompositions of cells and spheres., Mathematica Scandinavica 29 (1971), 197–205.1 [McM70] P. McMullen, The maximum numbers of faces of a convex polytope, Mathematika 17 (1970), 179–184. MR 0283691 (44 #921)4, 4.2 [McM71] , The numbers of faces of simplicial polytopes, Israel J. Math. 9 (1971), 559–570. MR 0278183 (43 #3914)4 [McM77] , Valuations and Euler-type relations on certain classes of convex polytopes, Proc. London Math. Soc. (3) 35 (1977), no. 1, 113–135. MR 0448239 (56 #6548) 5.2, 5.3 [McM93] Peter McMullen, On simple polytopes, Invent. Math. 113 (1993), no. 2, 419–444. MR 1228132 (94d:52015) 5.2 [Mot57] Theodore S Motzkin, Comonotone curves and polyhedra, Bull. Am. Math. Soc 63 (1957), 35.4, 4.2 [PK92] A. V. Pukhlikov and A. G. Khovanski˘ı, The Riemann-Roch theorem for integrals and sums of quasipolynomials on virtual polytopes, Algebra i Analiz 4 (1992), no. 4, 188–216. MR 1190788 (94c:14044) 5.3 [Poi93] Henri Poincar´e, Sur la g´en´eralisation dun th´eoreme deuler relatif aux polyedres, Comptes Rendus de S´eancesde lAcademie des Sciences 117 (1893), 144.1 [Rei76] Gerald Allen Reisner, Cohen-Macaulay quotients of polynomial rings, Advances in Math. 21 (1976), no. 1, 30–49. MR 0407036 (53 #10819)4 [Sch50] Ludwig Schl¨afli, Theorie der vielfachen kontinuit¨at, Gesammelte Mathematische Abhandlungen, Springer Basel, 1950, pp. 167–387 (German).1 [Sch59] Marcel-Paul Sch¨utzenberger, A characteristic property of certain polynomials of ef moore and ce shannon, RLE Quaterly Progress Report 55 (1959), 117–118.3, 3.2 [Som27] D. M. Y. Sommerville, The relations connecting the angle-sums and volume of a polytope in space of n dimensions, Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 115 (1927).2, 2.1 [Sta75] Richard P. Stanley, The upper bound conjecture and Cohen-Macaulay rings, Studies in Appl. Math. 54 (1975), no. 2, 135–142. MR 0458437 (56 #16640)4, 4.2 [Sta80] , The number of faces of a simplicial convex polytope, Adv. in Math. 35 (1980), no. 3, 236–238. MR 563925 (81f:52014)4 [Sta94] , Flag f-vectors and the cd-index, Math. Z. 216 (1994), no. 3, 483–499. MR 1283084 (96b:06006) 2.3 [Sta96] , Combinatorics and commutative algebra, second ed., Progress in Mathematics, vol. 41, Birkh¨auserBoston, Inc., Boston, MA, 1996. MR 1453579 (98h:05001) 4.1 12 AHMED UMER ASHRAF

[Sta12] , Enumerative combinatorics. Volume 1, second ed., Cambridge Studies in Advanced Mathe- matics, vol. 49, Cambridge University Press, Cambridge, 2012. MR 2868112 2.1 [Ste10] Ernst. Steinitz, Polyeder und raumeinteilungen, Encyklop¨adieder mathematischen Wissenschaften mit Einschluss ihrer Anwendungen, vol. 3 (Geometrie), Teubner, 1910, pp. 1–139.1 [Tim99] V. A. Timorin, An analogue of the Hodge-Riemann relations for simple convex polyhedra, Uspekhi Mat. Nauk 54 (1999), no. 2(326), 113–162. MR 1711255 (2001b:52018) 5.3, 5.7