THE VERONESE CONSTRUCTION FOR FORMAL POWER SERIES AND GRADED ALGEBRAS

FRANCESCO BRENTI AND VOLKMAR WELKER

Abstract. Let (a_{n})_{n≥0} be a sequence of complex numbers such that its generating series satisfies ∑_{n≥0} a_{n}t^{n} = \frac{h(t)}{(1−t)^{d}} for some polynomial h(t). For any r ≥ 1 we study the transformation of the coefficient series of h(t) to that of h^{⟨r⟩}(t) where ∑_{n≥0} a_{nr}t^{n} = \frac{h^{⟨r⟩}(t)}{(1−t)^{d}}. We give a precise description of this transformation and show that under some natural mild hypotheses the roots of h^{⟨r⟩}(t) converge when r goes to infinity. In particular, this holds if ∑_{n≥0} a_{n}t^{n} is the Hilbert series of a standard graded k-algebra A. If in addition A is Cohen-Macaulay then the coefficients of h^{⟨r⟩}(t) are monotonely increasing with r. If A is the Stanley-Reisner ring of a simplicial complex Δ then this relates to the rth edgewise subdivision of Δ which in turn allows some corollaries on the behavior of the respective f-vectors.

1. Introduction and Statement of Results

In this paper we study for a rational formal power series of the form

f(t) := ∑_{n≥0} a_{n}t^{n} = \frac{h(t)}{(1 − t)^{d}} , a_{n} ∈ ℂ for n ≥ 0,

the transformation of the numerator polynomial when passing for some number r ≥ 1 to the generating function f^{⟨r⟩}(t) := ∑_{n≥0} a_{rn}t^{n} = \frac{h^{⟨r⟩}(t)}{(1−t)^{d}}.

We are motivated by the following facts from commutative algebra. Let A = ⨁_{n≥0} A_{n} be a standard graded k-algebra; that is A is finitely generated in degree 1 and A_{0} = k. The Hilbert-Serre Theorem [10, see Chap. 10.4] asserts that its Hilbert-series Hilb(A, t) = ∑_{n≥0} dim_{k} A_{n} t^{n} is a rational function of the form Hilb(A, t) = \frac{h(t)}{(1−t)^{d}} for some polynomial h(t) such that h(1) ≥ 1 and d the Krull dimension of A. The rth Veronese algebra of A is the standard graded k-algebra A^{⟨r⟩} := ⨁_{n≥0} A_{nr} with Hilbert-series Hilb(A^{⟨r⟩}, t) = ∑_{n≥0} dim_{k} A_{rn}t^{n} = Hilb(A, t)^{⟨r⟩}. Veronese algebras are well studied objects in commutative algebra and algebraic geometry. In particular, the limiting behavior of algebraic properties of A^{⟨r⟩} for large r has been considered in [1], [9] and more generally in [11]. Indeed as will be seen later the results from [1] and [9] relate to our own results.

Key words and phrases. Veronese algebra, Hilbert series, h-vector, real-rootedness, unimodality, edgewise subdivision.

Both authors were partially supported by Ateneo Italo-Tedesco and DAAD through a Vigoni project.

Most results presented in this paper are very much in the spirit of results from [4] on the behavior of the h-vector and h-polynomial of barycentrically subdivided simplicial complexes. But even though the formulations of the theorems appear to be very similar the proofs are almost disjoint except for the use of Lemma 3.3 in the proof of Theorem 1.4.

Before we can state the first main result we have to define the following numbers. Here and in the sequel ℕ will denote the nonnegative integers and ℙ the strictly positive integers. For d, r, i ∈ ℕ let

C(r, d, i) := |\{ (a_{1}, …, a_{d}) ∈ ℕ^{d} | \begin{matrix}a_{1} + ⋯ + a_{d} = i \\ a_{j} ≤ r for 1 ≤ j ≤ d \end{matrix} \}|,

for d ≥ 1 and C(r, 0, i) = δ_{0,i}. Note that it is easy to see that

C(r, d, i) = ∑_{\{λ⊆(r^{d}): |λ|=i\}} (\begin{matrix}d \\ m_{1}(λ),...,m_{r}(λ),d−l(λ) \end{matrix}) ,

where, for a partition λ, ℓ(λ) denotes the number of parts of λ and m_{i}(λ) the number of parts of λ that are equal to i.

Theorem 1.1. Let (a_{n})_{n≥0} be a sequence of complex numbers such that for some s, d ≥ 0 its generating series f(t) := ∑_{n≥0} a_{n}t^{n} satisfies

f(t) = \frac{h_{0} + ⋯ + h_{s}t^{s}}{(1 − t)^{d}} .

Then for any r ∈ ℙ we have

f^{⟨r⟩}(t) = ∑_{n≥0} a_{nr}t^{n} = \frac{h_{0}^{⟨r⟩} + ⋯ + h_{m}^{⟨r⟩}t^{m}}{(1 − t)^{d}} ,

where m:=max(s,d) and

h_{i}^{⟨r⟩} = ∑_{j=0}^{s} C(r − 1, d, ir − j) h_{j},

for i = 0, …, m.

The following is a simple reformulation of Theorem 1.1 in the case of Hilbertseries of standard graded k-algebras.

Corollary 1.2. Let A be a standard graded k-algebra of dimension d with Hilbertseries

Hilb(A, t) = \frac{h_{0} + ⋯ + h_{s}t^{s}}{(1 − t)^{d}} .

Then for any r ∈ ℙ we have

Hilb(A^{⟨r⟩}, t) = \frac{h_{0}^{⟨r⟩} + ⋯ + h_{m}^{⟨r⟩}t^{m}}{(1 − t)^{d}} ,

where m:=max(s,d) and

h_{i}^{⟨r⟩} = ∑_{j=0}^{s} C(r − 1, d, ir − j) h_{j},

for i = 0, …, m.

As a consequence it follows that the first d + 1 entries of the h-vector h(A) = (h_{0}, …, h_{s}) of a standard graded algebra grow weakly when taking Veronese subalgebras in case h_{i} ≥ 0 for 0 ≤ i ≤ s. Note that this condition for example is satisfied if A is Cohen-Macaulay (see for example [2, Proposition 4.3.1]).

Corollary 1.3. Let A be a standard graded k-algebra of dimension d with Hilbertseries

Hilb(A, t) = \frac{h_{0} + ⋯ + h_{s}t^{s}}{(1 − t)^{d}}

such that h_{i} ≥ 0 for i = 0, …, s. Then for any r ≥ 1 the Hilbert-series

Hilb(A^{⟨r⟩}, t) = \frac{h_{0}^{⟨r⟩} + ⋯ + h_{m}^{⟨r⟩}t^{m}}{(1 − t)^{d}}

of the rth Veronese algebra of A satisfies h_{i}^{⟨r⟩} ≥ h_{i} for 0 ≤ i ≤ d. Moreover, if r ≥ d then h_{i}^{⟨r⟩} > h_{i} for 1 ≤ i ≤ d − 1. In particular, all conclusions hold if A is Cohen-Macaulay.

Proof. By Corollary 1.2 it follows that h_{i}^{⟨r⟩} = ∑_{j=0}^{s} C(r − 1, d, ir − j) h_{j}. Clearly, C(r − 1, d, ir − j) ≥ 0 for all r, d, i, j. Moreover, for 0 ≤ i ≤ d we have C(r − 1, d, ir − i) ≥ 1 since the sum

i(r − 1) = \underbrace{(r − 1) + ⋯ + (r − 1)}_{i times} + \underbrace{0 + ⋯ + 0}_{(d−i) times}

is clearly among the sums counted by C(r − 1, d, i(r − 1)). This implies h_{i}^{⟨r⟩} ≥ h_{i} for 0 ≤ i ≤ d. It is well known that for a standard graded algebra we have h_{0} = 1. Now if r ≥ d then for 1 ≤ i ≤ d − 1 we have ir ≤ (d − 1)r ≤ d(r − 1). Thus there is at least one sum representation of ir with d summands ≤ r − 1. Hence C(r − 1, d, ir) ≥ 1 and therefore C(r − 1, d, ir − 0)h_{0} ≥ 1 which then implies h_{i}^{⟨r⟩} ≥ h_{i} + 1 for 1 ≤ i ≤ d − 1. □

Note that for the Hilbert-series Hilb(A, t) = \frac{h_{0}+···+h_{d}t^{d}}{(1−t)^{d}} of a standard graded

algebra A it is well known that h_{0}^{⟨r⟩} = h_{0} = 1 and h_{d}^{⟨r⟩} = h_{d}. Of course these identities also follow from Theorem 1.1.

4 FRANCESCO BRENTI AND VOLKMAR WELKER

Theorem 1.4. For any d ≥ 2 there are strictly negative real numbers α_{1} …, α_{d−2} such that for any sequence (a_{n})_{n≥0} of real numbers such that a_{0} = 1 and a_{n} ≥ 0 for large n whose generating series f(t) = ∑_{n≥0} a_{n}t^{n} = \frac{h(t)}{(1−t)^{d}} for some polynomial h(t) with h(1) ≠ 0 there is a number R > 0 and sequences of complex numbers (β_{r}^{(i)})_{r≥1}, 1 ≤ i ≤ d, such that :

(i) β_{r}^{(i)} is real for r > R and 1 ≤ i ≤ d and strictly negative for r > R and 1 ≤ i ≤ d − 1.

(ii) β_{r}^{(i)} → α_{i} for r → ∞ and 1 ≤ i ≤ d − 2.

(iii) β_{r}^{(d−1)} → −∞ for r → ∞.

(iv) β_{r}^{(d)} → 0 for r → ∞.

(v) h_{0}^{⟨r⟩} + ⋯ + h_{d}^{⟨r⟩}t^{d} = ∏_{i=1}^{d}(1 − β_{r}^{(i)}t).

Corollary 1.5. For any d ≥ 2 there are strictly negative real numbers α_{1} …, α_{d−2} such that for any standard graded k-algebra of dimension d with Hilbert-series

Hilb(A, t) = \frac{h_{0} + ⋯ + h_{s}t^{s}}{(1 − t)^{d}}

there are R > 0 and sequences of complex numbers (β_{r}^{(i)})_{r≥1}, 1 ≤ i ≤ d, such that :

(i) β_{r}^{(i)} is real for r > R and 1 ≤ i ≤ d and strictly negative for r > R and 1 ≤ i ≤ d − 1.

(ii) β_{r}^{(i)} → α_{i} for r → ∞ and 1 ≤ i ≤ d − 2.

(iii) β_{r}^{(d−1)} → −∞ for r → ∞.

(iv) β_{r}^{(d)} → 0 for r → ∞.

(v) h_{0}^{⟨r⟩} + ⋯ + h_{d}^{⟨r⟩}t^{d} = ∏_{i=1}^{d}(1 − β_{r}^{(i)}t), for r > R.

In [1] and [9] it is shown that for a standard graded k-algebra A and r large enough the rth Veronese A^{⟨r⟩} is Koszul (see [10, p. 450]). In turn it is known (see for example [13]) that this property implies that the numerator polynomial of the Hilbert-series has at least one real root. Thus in some sense the previous and the following corollary are inspired by this algebraic limiting results.

Corollary 1.6. Let A be a standard graded k-algebra of dimension d ≥ 1 with Hilbert-series

Hilb(A, t) = \frac{h_{0} + ⋯ + h_{s}t^{s}}{(1 − t)^{d}} .

Then there is R > 0 such that for any r > R we have:

(i) h_{i}^{⟨r⟩} ≥ 1, 0 ≤ i ≤ d − 1,

(ii) h_{i}^{⟨r⟩} = 0 for i ≥ d + 1

(iii) h^{⟨r⟩}(t) has only real zeros. In particular, for r > R the sequence (h_{0}^{⟨r⟩}, …, h_{d}^{⟨r⟩}) is log-concave and unimodal.

FORMAL POWER SERIES AND ALGEBRAS 5

2. Proof of Theorem 1.1

Proof of Theorem 1.1. Set h(t) := ∑_{i=0}^{s} h_{i} t^{i} so that f(t) = \frac{h(t)}{(1−t)^{d}}. Let ρ ∈ ℂ be a primitive r-th root of unity. Then ρ and all its powers ρ^{j} for any j ∈ ℕ such that j ≢ 0 (mod r) satisfy ∑_{i=0}^{r−1}(ρ^{j})^{i} = 0. Hence:

∑_{n≥0} a_{rn}t^{rn} = \frac{1}{r} ∑_{i=0}^{r−1} f(ρ^{i} t)

= \frac{1}{r} ∑_{i=0}^{r−1} \frac{h(ρ^{i} t)}{(1 − ρ^{i} t)^{d}}

= \frac{∑_{i=0}^{r−1} (\frac{1−t^{r}}{1−ρ^{i} t})^{d} h(ρ^{i} t)}{r(1 − t^{r})^{d}}

= \frac{∑_{i=0}^{r−1} (\frac{1−(ρ^{i} t)^{r}}{1−ρ^{i} t} )^{d} h(ρ^{i} t)}{r (1 − t^{r})^{d}}

But

\frac{1}{r} ∑_{ℓ=0}^{r−1} (\frac{1 − (ρ^{ℓ}t)^{r}}{1 − ρ^{ℓ}t} )^{d} h(ρ^{ℓ}t) = \frac{1}{r} ∑_{ℓ=0}^{r−1} ((1 + ρ^{ℓ} t + ⋯ + (ρ^{ℓ} t)^{r−1})^{d}h(ρ^{ℓ} t))
= \frac{1}{r} ∑_{ℓ=0}^{r−1} (∑_{i≥0} C(r − 1, d, i)(ρ^{ℓ} t)^{i}) h(ρ^{ℓ} t)

= \frac{1}{r} ∑_{i≥0} C(r − 1, d, i) (∑_{ℓ=0}^{r−1}(ρ^{ℓ} t)^{i} ∑_{j=0}^{s} h_{j} (ρ^{ℓ} t)^{j})

= \frac{1}{r} ∑_{j=0}^{s} ∑_{i≥0} C(r − 1, d, i − j) h_{j} ∑_{ℓ=0}^{r−1}(ρ^{ℓ} t)^{i}

= ∑_{i≥0} (∑_{j=0}^{s} C(r − 1, d, ir − j) h_{j}) t^{ri}

and the result follows since if i > s ≥ d then ir − j > sr − s ≥ d(r − 1) for all 0 ≤ j ≤ s while if i > d > s then ir − j > dr − d for all 0 ≤ j ≤ s. □

We now analyze the transformation described in Theorem 1.1 more closely. Consider the vectorspace R_{d} of all rational functions \frac{h(t)}{(1−t)^{d}} for polynomials h(t) of degree ≤ d. We consider two bases of R_{d}. First the basis B_{d}^{1} consisting of all \frac{t^{i}}{(1−t)^{d}} for 0 ≤ i ≤ d. For the second basis we recall the definition of an Eulerian polynomial. For a number i ≥ 1 we define A_{i}(t) = ∑_{σ∈S_{i}} t^{des(σ)+1}, where des(σ) is the number of descents of the permutation σ. If we also set A_{0}(t) = A_{−1}(t) = 1 then the set B_{d}^{2} consisting of all \frac{A_{i−1}(t)(1−t)^{d−i}}{(1−t)^{d}}, 0 ≤ i ≤ d, is a second basis of R_{d}. Using the fact that for 1 ≤ i one has A_{i}(1) ≠ 0 one easily checks that indeed B_{d}^{2} is a basis of R_{d}.

For a fixed r ≥ 1 let Φ_{r} : R_{d} → R_{d} be the map that sends f(t) to f^{⟨r⟩}(t). Note that in the basis B_{d}^{2} the map Φ_{r} sends ∑_{n≥0} n^{i}t^{n} = \frac{A_{i}(t)}{(1−t)^{i+1}} for 0 ≤ i < d to

Φ_{r}(∑_{n≥0} n^{i}t^{n}) = ∑_{n≥0}(rn)^{i}t^{n}
= r^{i} ∑_{n≥0} n^{i}t^{n}

= r^{i} \frac{A_{i}(t)}{(1 − t)^{i+1}}

while Φ_{r}(A_{−1}(t)) = A_{−1}(t). In particular, this confirms that Φ_{r} is a map from R_{d} to R_{d}. Moreover, since Φ_{r} is easily seen to be linear it follows that in the basis B_{d}^{2} the map Φ_{r} is represented by the (d + 1) × (d + 1) diagonal matrix diag(1, 1, r, …, r^{d−1}). The preceding arguments and Theorem 1.1 imply the following lemma.

Lemma 2.1. Let ℭ_{d,r} = (C(r − 1, d, ir − j))_{0≤i,j≤d}. Then ℭ_{d,r} is the matrix representing the linear transformation Φ_{r} with respect to the basis B_{d}^{1}. In particular, ℭ_{d,r} is diagonizable with eigenvalues 1 of multiplicity two and r, …, r^{d−1} of multiplicity one.

Indeed we can give a factorization of ℭ_{d,r} which also clarifies its eigenspaces. Let 𝔏_{d} = (l_{ij})_{0≤i,j≤d} be the (d + 1) × (d + 1) matrix with entries l_{ij} defined by A_{i−1}(t)(1 − t)^{d−i} = ∑_{j=0}^{d} l_{j,i}t^{j}.

Lemma 2.2. For any d, r ≥ 1 we have

ℭ_{d,r} = 𝔏_{d} · diag(1, 1, r, …, r^{d−1}) · 𝔏_{d}^{−1}.

The vector ℓ_{i} = (l_{0i}, …, l_{di})^{t} is an eigenvector of ℭ_{d,r} for the eigenvalue r^{i−1} for 1 ≤ i ≤ d. Moreover:

(i) l_{d,i} = 0 for 1 ≤ i ≤ d and l_{d,0} = (−1)^{d}.

(ii) l_{0,i} = 0 for 2 ≤ i ≤ d and l_{0,0} = l_{0,1} = 1.

(iii) l_{j,d} ≥ 1, 1 ≤ j ≤ d − 1.

Proof. The matrix 𝔏_{d} describes the base change from B_{d}^{2} to B_{d}^{1}. Since by the arguments preceding Lemma 2.1 diag(1, 1, r, …, r^{d−1}) is the matrix of Φ_{r} with respect to the basis B_{d}^{2} and since by Theorem 1.1 ℭ_{d,r} is the matrix of Φ_{r} with respect to the basis B_{d}^{1} it follows that ℭ_{d,r} = 𝔏_{d}diag(1, 1, r, …, r^{d−1})𝔏_{d}^{−1}. The preimage of the ith unit vector under 𝔏_{d}^{−1} is ℓ_{i}. Since the ith unit vector is an eigenvector for r^{i−1} of diag(1, 1, r, …, r^{d−1}) it follows by the first part of the lemma that ℓ_{i} is an eigenvector of ℭ_{d,r} for the eigenvalue r^{i−1}, 1 ≤ i ≤ d.

Assertions (i) - (iii) are immediate consequences of the definitions. □

The preceding lemma implies the following combinatorial identity for Eulerian numbers. For d ≥ 1 and 0 ≤ i ≤ d let A(d, i) = |{σ ∈ S_{d} : des(σ) = i − 1}|.

Proposition 2.3. Let d, r ≥ 1. Then

∑_{j=0}^{d} C(r − 1, d + 1, ir − j)A(d, j) = r^{d}A(d, i)

for i = 0, …, d. In particular

∑_{j=0}^{d} C(r − 1, d + 1, r − j)A(d, j) = r^{d}.

Clearly, Proposition 2.3 asks for a combinatorial proof.

3. Proof of Theorem 1.4

Before we come to the proof of Theorem 1.4 we need some preparatory lemmas.

Lemma 3.1. Let f(t) = \frac{h_{0}+···+h_{s}t^{s}}{(1−t)^{d}} for some d, s ≥ 0. Write f(t) = p(t) + \frac{h_{1}(t)}{(1−t)^{d}} for polynomials p(t) and h_{1}(t) where h_{1}(t) is of degree ≤ d − 1. Then for any r ≥ s − d + 1 we have

f(t)^{⟨r⟩} = \frac{h^{⟨r⟩}(t)}{(1 − t)^{d}} = f_{1}(t)^{⟨r⟩}

for some polynomial h^{⟨r⟩}(t) of degree ≤ d and f_{1}(t) = \frac{p(0)(1−t)^{d}+h_{1}(t)}{(1−t)^{d}}. Moreover, h_{0} = h^{⟨r⟩}(0) for all r.

Proof. If s ≤ d then the result follows immediately from Theorem 1.1 so assume s > d. Then f(t) = p_{0} + p_{1}t + ⋯ + p_{s−d}t^{s−d} + \frac{h_{1}(t)}{(1−t)^{d}} for a polynomial h_{1}(t) of

degree ≤ d − 1. Hence f(t)^{⟨r⟩} = p_{0} + \frac{h_{1}(t)^{⟨r⟩}}{(1−t)^{d}} = f_{1}(t)^{⟨r⟩} for r ≥ s − d + 1. Since by Theorem 1.1 h_{1}(t)^{⟨r⟩} is a polynomial of degree ≤ d the assertion follows.

The last equality follows by evaluating the generating series at t = 0. □

Remark 3.2. Let f(t) = ∑_{n=0}^{∞} a_{n}t^{n} = \frac{h(t)}{(1−t)^{d}}. If d = 0 then for sufficiently large r we have f(t)^{⟨r⟩} = a_{0}. If d = 1 then for sufficiently large r we have f(t)^{⟨r⟩} = \frac{h_{0}+h_{1}t}{1−t} with h_{0}, h_{1} independent of r. In particular, the numerator polynomial of f(t)^{⟨r⟩} is real rooted with at most one root which is independent of r.

We recall the following lemma from [4].

Lemma 3.3 (Lemma 4.9 [4]). Let (g_{n}(t))_{n≥0} be a sequence of real polynomials of degree d − 2, f(t) another real polynomial of degree d − 2 and ρ > 1, h_{d} real numbers such that:

▷ lim_{n→∞} g_{n}(t)/ρ^{n} = 0, where the limit is taken in ℝ^{d−1}.

▷ All the roots of the polynomial f(t) are strictly negative and all coefficients of f(t) are strictly positive.

Then there are real numbers α_{i}, 1 ≤ i ≤ d − 2 and sequences (β_{i}^{(n)})_{n≥0}, 1 ≤ i ≤ d of complex numbers such that:

(i) β_{i}^{(n)}, 1 ≤ i ≤ d, are real for n sufficiently large.

(ii) lim_{n→∞} β_{i}^{(n)} = α_{i}, 1 ≤ i ≤ d − 2.

(iii) lim_{n→∞} β_{d−1}^{(n)} = 0.

(iv) lim_{n→∞} β_{d}^{(n)} = −∞.

(v) ∏_{i=0}^{d−1}(t − β_{i}^{(n)}) = h_{d} + tg_{n}(t) + ρ^{n}tf(t) + t^{d}.

Proof of Theorem 1.4. By Lemma 3.1 we may assume that h(t) is of degree ≤ d. Then Theorem 1.1 implies that for f^{⟨r⟩}(t) = \frac{h^{⟨r⟩}(t)}{(1−t)^{d}} the polynomial h^{⟨r⟩}(t) is again of degree ≤ d.

Let ℓ_{i} = (l_{0i}, …, l_{d,i})^{t}, 0 ≤ i ≤ d be the eigenvectors of ℭ_{d,r} as in Lemma

2.2. Let ℓ_{i}(t) = ^{\frac{∑_{j=0}^{d} l_{j,i}t^{j}}{(1−t)^{d}}}, 0 ≤ i ≤ d so B_{d}^{2} = \{ℓ_{0}(t), …, ℓ_{d}(t)\}. Let f(t) = α_{0}ℓ_{0}(t) + ⋯ + α_{d}ℓ_{d}(t) be the expansion of f(t) in the basis B_{d}^{2}. Then by Lemma 2.2 we have f(t)^{⟨r⟩} = α_{0}ℓ_{0}(t) + α_{1}ℓ_{1}(t) + α_{2}rℓ_{2}(t) + ⋯ + α_{d}r^{d−1}ℓ_{d}(t). Let

g_{r}(t) =^{def} t^{d} h^{⟨r⟩} (\frac{1}{t}) − h_{d} − t^{d} − r^{d−1} α_{d} t^{d} ℓ̃_{d} (\frac{1}{t})

where ℓ̃_{d}(t) =^{def} (1−t)^{d} ℓ_{d}(t). Then deg(g_{r}) ≤ d−1, g_{r}(0) = 0 and lim_{r→∞} g_{r}(t)/r^{d−1} = 0. Now ℓ̃_{d}(t) = A_{d−1}(t) is the (d − 1)st Eulerian polynomial. Thus ℓ_{d}(t) = t^{d}ℓ_{d}(1/t). Hence if we set g̃_{r}(t) =^{def} g_{r}(t)/t, f̃(t) =^{def} A_{d−1}(t)/t then f̃(t) is a polynomial of degree d − 2, which is real rooted by [6, p. 292, Ex. 3] and has strictly positive coefficients except for the constant, and all the hypotheses of Lemma 3.3 are satisfied.

Hence Lemma 3.3 becomes applicable and the result follows by passing to the reciprocal polynomials. □

It remains to provide proofs of Corollaries 1.5 and 1.6. Corollary 1.5 is just a reformulation of Theorem 1.4 for Hilbert-series. So there is nothing to prove. But note that in general passing to Veronese subrings A^{⟨r⟩} for small r does not suffice to guarantee that h^{⟨r⟩}(t) is real rooted, even in case h(t) is of degree d−1 and has strictly positive coefficients. For example, f(t) = \frac{1+t+t^{2}+t^{3}+t^{4}}{(1−t)^{5}} is the Hilbert-series of the Stanley-Reisner ring of the boundary complex of the 4-simplex. However f^{⟨2⟩}(t) = \frac{1+16t+31t^{2}+26t^{3}+6t^{4}}{(1−t)^{5}} whose numerator polynomial has only two real roots.

Proof of 1.6. If d = 1 then (iii) follows from (ii) which follows immediately from Lemma 3.1.

Let d ≥ 2. Assertions (ii) and (iii) follow immediately from Lemma 3.1 and Corollary 1.5. From the proof of Theorem 1.4 we recall that the numerator polynomial of Hilb(A^{⟨r⟩}, t) can be written as

(3.1)  h_{d}t^{d} + t^{d}g_{r} (\frac{1}{t}) + α_{d}r^{d−1}A_{d−1}(t) + 1

for polynomials g_{r}(t) and the Eulerian polynomial A_{d−1}(t). Moreover we have lim_{r→∞} g_{r}(t)/r^{d−1} = 0 and A_{d−1}(t) has strictly positive coefficients except for the constant term which is 0. But this implies that for large enough r the polynomial 1/r^{d−1}(t^{d}g_{r}(1/t)+r^{d−1}A_{d−1}(t)) has strictly positive coefficients. This then implies that for large enough r the polynomial in (3.1) has strictly positive coefficients except for possibly h_{d}, which proves (i). □

4. Veronese of Stanley-Reisner rings

For a simplicial complex Δ over ground set Ω and a field k we denote by k[Δ] its Stanley-Reisner ring. Recall that k[Δ] = k[x_{ω} : ω ∈ Ω]/I_{Δ}, where I_{Δ} is the ideal generated by the monomials ∏_{ω∈A} x_{ω} for A ∉ Δ. Assume that Δ is (d − 1)-dimensional. We denote by f(Δ) = (f_{−1}, …, f_{d−1}) the f-vector of Δ; that is f_{i} is the number of i-dimensional faces of Δ. Then it is well known that :

(4.1)  Hilb(k[Δ], t) = \frac{h_{0} + ⋯ + h_{d}t^{d}}{(1 − t)^{d}}

= \frac{∑_{i=0}^{d} f_{i−1}t^{i}(1 − t)^{d−i}}{(1 − t)^{d}}

The rth Veronese of k[Δ] is a Stanley-Reisner ring only in extremal cases, but still it has turned out to be fruitful and meaningful to look for a simplicial complex Δ^{⟨r⟩} such that

(4.2)  Hilb(k[Δ]^{⟨r⟩}, t) = Hilb(k[Δ^{⟨r⟩}], t).

In [3] based on earlier ideas by Sturmfels [15], Brun and Römer consider the following situation. Set S = k[x_{1}, …, x_{n}] the polynomial ring and let I_{Δ} be the Stanley-Reisner ideal of a simplicial complex Δ on ground set [n]. Then the r-th Veronese (S/I_{Δ})^{⟨r⟩} can be described as a quotient of the polynomial ring S(r) in the variables x_{i_{1},...,i_{n}} indexed by numbers 0 ≤ i_{1}, …, i_{n} such that i_{1} +⋯+i_{n} = r. If I(r) is the ideal in S(r) such that (S/I_{Δ})^{⟨r⟩} = S(r)/I(r) then Brun and Römer [3, Section 6] describe a initial ideal of I(r) which is the Stanley-Reisner ideal of a simplicial complex Δ(r) on vertex set Ω_{r} = {(i_{1}, …, i_{n}) ∈ ℕ^{n} : i_{1} +⋯+i_{n} = r}. By basic facts about initial ideals it follows that this Δ(r) fulfills (4.2). The simplicial complex Δ(r) turns out to be realizable as a subdivision of Δ. This subdivision is called r-th edgewise subdivision and as outlined in [3] has a long history in algebraic topology (see e.g. [8], [12]) and a shorter one in discrete geometry (see e.g.[7]).

Before we can describe edgewise subdivision we need some technical preparations. Consider ℝ^{n} together with its standard unit basis vectors 𝔢_{1}, …, 𝔢_{n}. By the obvious identification we can consider Δ as a simplicial complex over the ground set {𝔢_{1}, …, 𝔢_{n}} = Ω_{1}. Note that for r ≥ 1 the elements of Ω_{r} are the points with integer coordinates in the the r-th dilation of the simplex spanned by Ω_{1}. For a vector 𝔞 = (a_{1}, …, a_{n}) ∈ ℝ^{n} its support supp(𝔞) is the set {i : a_{i} ≠ 0} of indices of non-zero coordinates. For i ∈ [n] set 𝔲_{i} = 𝔢_{i} + 𝔢_{i+1} + ⋯ + 𝔢_{n} and for 𝔞 = (a_{1}, …, a_{n}) ∈ ℝ^{n} set ι(𝔞) := ∑_{i=1}^{n} a_{i}𝔲_{i}.

The r-th edgewise subdivision of Δ is the simplicial complex Δ(r) on ground set Ω_{r} such that A ⊆ Ω_{r} is a simplex of Δ(r) if and only if

(ESD1) ⋃_{𝔳∈A}supp(𝔳) ∈ Δ.

(ESD2) For all 𝔳, 𝔳′ ∈ A either ι(𝔳 − 𝔳′) ∈ {0, 1}^{n} or ι(𝔳′ − 𝔳) ∈ {0, 1}^{n}.

Now the result by Brun and Römer [3] states.

Proposition 4.1 (Proposition 6.4 in [3]). Let Δ be a simplicial complex on ground set [n] and I(r) be such that (S/I_{Δ})^{⟨r⟩} = S(r)/I(r). Then there is a term order for which I_{Δ(r)} is the initial ideal of I(r).

In particular, Δ^{⟨r⟩} := Δ(r) satisfies Equation (4.2). Clearly, a simplicial complex Δ^{⟨r⟩} satisfying Equation (4.2) is not uniquely defined. But Δ(r) appears to be a natural choice.

First we want to study the enumerative properties of Δ(r). A partial analysis can also be found in [7] but there the final formulas appear as alternating sums which is not fully satisfactory from the point of view of Enumerative Combinatorics. For enumerative purposes Equation (4.1) suggests to study a third basis of R_{d}. We denote by B_{d}^{3} the set of rational functions \frac{t^{i}(1−t)^{d−i}}{(1−t)^{d}}, 0 ≤ i ≤ d. Indeed B_{d}^{3} will be crucial in the proof of the following proposition.

Proposition 4.2. Let Δ be a simplicial complex of dimension d − 1 with fvector f(Δ) = (f_{−1}, …, f_{d−1}) and r ∈ ℙ. If Δ^{⟨r⟩} is a simplicial complex such that Hilb(k[Δ]^{⟨r⟩}, t) = Hilb(k[Δ^{⟨r⟩}], t) then its f-vector f(Δ^{⟨r⟩}) = (f_{−1}^{⟨r⟩}, …, f_{d−1}^{⟨r⟩}) satisfies

f_{i−1}^{⟨r⟩} = ∑_{ℓ=i}^{d} ∑_{{j_{1}+···+j_{i}=ℓ \atop j_{1},...,j_{i}≥1}} \binom{r − 1}{j_{1} − 1}\binom{r}{j_{2}} ⋯ \binom{r}{j_{i}}f_{ℓ−1},

for 0 ≤ i ≤ d.

Proof. We denote by v_{i}(t) = \frac{t^{i}(1−t)^{d−i}}{(1−t)^{d}}, 0 ≤ i ≤ d, the elements of B_{d}^{3}. Then there

are numbers a_{i,ℓ}^{(r)} (0 ≤ i, ℓ ≤ d) such that

(4.3)  v_{i}(t)^{⟨r⟩} = ∑_{ℓ=0}^{d} a_{i,ℓ}^{(r)} v_{ℓ}(t).

Hence

(4.4)  ∑_{i=0}^{∞} v_{i}(t)^{⟨r⟩}x^{i} = ∑_{i=0}^{∞} ∑_{ℓ=0}^{d} a_{i,ℓ}^{(r)} \frac{x^{i}t^{ℓ}}{(1 − t)^{ℓ}}

(4.5)  = ∑_{ℓ=0}^{d} ( ∑_{i=0}^{∞} a_{i,ℓ}^{(r)}x^{i})\frac{t^{ℓ}}{(1 − t)^{ℓ}}

Next we derive a second expansion of the left hand side of (4.4). First we derive an expansion of v_{i}(t)^{⟨r⟩} as a formal power series. Clearly, for i ≥ 1,

v_{i}(t) = \frac{t^{i}}{(1 − t)^{i}} = ∑_{j=0}^{∞} \binom{j − 1}{i − 1}t^{j}

Therefore, for i ≥ 1,

v_{i}(t)^{⟨r⟩} = ∑_{j=0}^{∞} \binom{r j − 1}{i − 1}t^{j}.

For i = 0 we have v_{0}^{⟨r⟩}(t) = 1 = v_{0}(t).

The preceding expansion leads to the following identity.

∑_{i=0}^{∞} v_{i}(t)^{⟨r⟩}x^{i} = 1 + ∑_{i=1}^{∞} ∑_{j=0}^{∞} \binom{r j − 1}{i − 1}t^{j}x^{i}

= 1 + ∑_{j=0}^{∞} (x ∑_{i=1}^{∞} \binom{r j − 1}{i − 1}x^{i−1})t^{j}

= 1 + ∑_{j=1}^{∞} x(1 + x)^{rj−1}t^{j}
(4.6)  = 1 + t(1 + x)^{r−1} \frac{x}{1 − t(1 + x)^{r}}

Writing the formulas from (4.5) and (4.6) in terms of the variable u = \frac{t}{1−t} or equivalently t = \frac{u}{1+u} and comparing we obtain

1 + u(1 + x)^{r−1} \frac{x}{1 + u − u(1 + x)^{r}} = ∑_{ℓ=0}^{d} ( ∑_{i=0}^{∞} a_{i,ℓ}^{(r)}x^{i})u^{ℓ}

Thus by

\frac{xu(1 + x)^{r−1}}{1 + u − u(1 + x)^{r}} = ∑_{ℓ=0}^{∞} x(1 + x)^{r−1}((1 + x)^{r} − 1)^{ℓ}u^{ℓ+1}

we obtain for ℓ ≥ 1

∑_{i=0}^{∞} a_{i,ℓ}^{(r)}x^{i} = x(1 + x)^{r−1}((1 + x)^{r} − 1)^{ℓ−1}

and ∑_{i=0}^{∞} a_{i,0}^{(r)} x^{i} = 1. Hence for any ℓ ≥ 0 and i ≥ 0

(4.7)  a_{i,ℓ}^{(r)} = ∑_{{j_{1}+···+j_{ℓ}=i \atop j_{1},...,j_{ℓ}≥1}} \binom{r − 1}{j_{1} − 1}\binom{r}{j_{2}} ⋯ \binom{r}{j_{ℓ}}

From this we conclude the following equalities which imply the assertion

∑_{i=0}^{d} f_{i−1}^{⟨r⟩} v_{i}(t) = Hilb(k[Δ^{⟨r⟩}], t) = Hilb(k[Δ]^{⟨r⟩}, t)
= Hilb(k[Δ], t)^{⟨r⟩} = ∑_{i=0}^{d} f_{i−1}v_{i}(t)^{⟨r⟩}
= ∑_{i=0}^{d} f_{i−1} ∑_{ℓ=0}^{d} a_{i,ℓ}^{(r)}v_{ℓ}(t)
= ∑_{ℓ=0}^{d}(∑_{i=0}^{d} f_{i−1}a_{i,ℓ}^{(r)}) v_{ℓ}(t)
= ∑_{ℓ=0}^{d} ( ∑_{i=ℓ}^{d} f_{i−1} ∑_{{j_{1}+···+j_{ℓ}=i \atop j_{1},...,j_{ℓ}≥1}} \binom{r − 1}{j_{1} − 1}\binom{r}{j_{2}} ⋯ \binom{r}{j_{ℓ}})·

The last equality follows from (4.7) and the fact that a_{i,ℓ}^{(r)} = 0 for i < ℓ. □ As an immediate consequence of Theorem 1.4 we also get the following result.

Proposition 4.3. For any d ≥ 2 there are real numbers α′_{1}, …, α′_{d−2} such that for any (d − 1)-dimensional simplicial complex Δ there are R > 0 and sequences of complex numbers (β_{r}^{(i)})_{r≥1}, 1 ≤ i ≤ d, such that :

(i) β_{r}^{(i)} is real for r > R and 1 ≤ i ≤ d and strictly negative for r > R and 1 ≤ i ≤ d − 1.

(ii) β_{r}^{(i)} → α′_{i} for r → ∞ and 1 ≤ i ≤ d − 2. (iii) β_{r}^{(d−1)} → −∞ for r → ∞, (iv) β_{r}^{(d)} → −1 for r → ∞,

(v) f_{−1}^{⟨r⟩} + ⋯ + f_{d−1}^{⟨r⟩}t^{d} = ∏_{i=1}^{d}(1 − β_{r}^{(i)}t), for r > R.

In particular, for r > R the f-vector of Δ^{⟨r⟩} is log-concave and unimodal.

In [4, Theorem 3.1] it is shown that for a simplicial complex with non-negative h-vector the h-polynomial of its first barycentric subdivision is real rooted. The example preceding the proof of Corollary 1.6 shows that such a result is not true for edgewise subdivision. More precisely, it shows that the h-polynomial of the second edgewise subdivision of the boundary of the 4-simplex is of degree 4 but has only two real roots.

Acknowledgment

We thank Aldo Conca and Bernd Sturmfels for suggesting the Veronese construction as an algebraic analog of barycentric subdivision of simplicial complexes. We also thank Aldo Conca and Tim Römer for providing references to edgewise subdivision and its relation to Veronese rings.

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Dipartimento di Matematica, Universita’ di Roma ”Tor Vergata”, Via della Ricerca Scientifica, 1, 00133, Roma, Italy

E-mail address: brenti@mat.uniroma2.it

Fachbereich Mathematik und Informatik, Philipps-Universität Marburg, 35032 Marburg, Germany

E-mail address: welker@mathematik.uni-marburg.de