J Novacoski

M Spivakovsky

ON STABLE AND FIXED POLYNOMIALS

Keywords: 2010 Mathematics Subject Classification. Primary 13A18 Key polynomials, stable polynomials, truncations of valuations, fixed polynomials

.

Introduction

Let ν be a rank one valuation on K[x]. For n ∈ N set

Ψ n = {Q ∈ K[x] | Q is a key polynomial for ν and deg(Q) = n}.
Suppose that Ψ n is non-empty and bounded (i.e., there exists a ∈ K[x] such that ν(a) > ν(Q) for every Q ∈ Ψ n ) and that ν(Ψ n ) does not have a maximum. Set

K[x] n = {a ∈ K[x] | deg(a) < n}. For each f ∈ K[x] and Q ∈ Ψ n the Q- expansion of f is the expression f = a 0 + a 1 Q + . . . + a r Q r = l(Q)
where l(X) ∈ K[x] n [X]. We denote the value r (which does not depend on the choice of Q ∈ Ψ n ) in the previous expression by deg X (f ). The truncation of ν on Q is given by ν

Q (f ) := min 0≤i≤r {ν(a i Q i )}. Let S n = {f ∈ K[x] | ν Q (f ) < ν(f ) for every Q ∈ Ψ n }.
A polynomial f is said to be Ψ n -stable if it does not belong to S n . A monic polynomial F ∈ K[x] is called a limit key polynomial for Ψ n if it belongs to S n and has the smallest degree among polynomials in S n .

In [START_REF] Kaplansky | Maximal fields with valuations I[END_REF], Kaplansky introduces the concept of pseudo-convergent sequences. These objects are strongly related to the set Ψ 1 . For a given such sequence a we can define what it means for a polynomial f ∈ K[x] to be fixed by a. Here, we generalize this concept for any of the sets Ψ n . We say that f = l(Q), l(X)

∈ K[x] n [X], is (Ψ n , Q)- fixed if there exists Q ′ ∈ Ψ n , ν(Q) < ν(Q ′ ) such that ν(f ) = ν(l(Q ′ -Q)).
Our first main result (Proposition 3.2) is that we can choose a suitable Q ∈ Ψ n such that for any f

∈ K[x] with deg X (f ) ≤ deg X (F ), we obtain that f is Ψ n -stable if and only if it is (Ψ n , Q)-fixed.
We will fix a suitable Q ∈ K[x] (see ( 2) and ( 3)). Then we fix a limit ordinal λ and any cofinal well-ordered (with respect to ν) subset

{Q ρ } ρ<λ ⊆ {Q ′ ∈ Ψ n | ǫ(Q) < ǫ(Q ′ )}. This means that if ρ < σ < λ, then ν(Q ρ ) < ν(Q σ ) and that for every Q ′ ∈ Ψ n , there exists ρ < λ such that ν(Q ′ ) < ν(Q ρ ). For each ρ < λ set h ρ := Q -Q ρ ∈ K[x] n and γ ρ = ν(Q ρ ).
It follows from the definition that {h ρ } ρ<λ is a pseudoconvergent sequence for ν. For simplicity, we will denote ν Qρ by ν ρ .

Let p = char(Kν) and I be the set of all non-negative powers of p. As an application of Proposition 3.2 we can prove the following.

Theorem 1.1. Let F be a limit key polynomial for Ψ n and write

F = L(Q) for some L(X) ∈ K[x] n [X].
Then we have the following.

(i): There exists σ < λ such that for every θ > σ the polynomial

F p = L(h θ ) + i∈I ∂ i L(h θ )Q i θ is a limit key polynomial for Ψ n . Here ∂ i L denotes the Hasse derivative of L(X) (as a polynomial in K(x)[X]) of order i. (ii): For each i ∈ I ∪ {0} there exists a i ∈ K[x] n such that F p = i∈I∪{0} a i Q i θ is a limit key polynomial for Ψ n .
Kaplansky proved the above result in the case n = 1. In that case, (ii) follows trivially from (i). Our proof of Theorem 1.1 follows Kaplansky's proof.

If n > 1, then (ii) was proven in [START_REF] Herrera Govantes | Key polynomials for simple extensions of valued fields[END_REF] (Proposition 8.2). An alternative proof of it was presented in [START_REF] Moraes | Limit key polynomials as p-polynomials[END_REF] (Theorem 1.2). The advantage of our proof is that it is much simpler and presents as algorithm on how to construct the limit key polynomial of this form (from a given limit key polynomial). Also, our proof does not require that are in the equicharacteristic case.

Preliminaries

Throughout this paper ν will denote a rank one valuation on K

[x]. For f ∈ K[x] we denote (1) ǫ(f ) = max 1≤b≤deg(f ) ν(f ) -ν(∂ b f ) b ,
where ∂ b f denotes the Hasse derivative of f of order b. We denote

I(f ) = b, 1 ≤ b ≤ deg(f ) | ǫ(f ) = ν(f ) -ν(∂ b f ) b .
A monic polynomial Q ∈ K[x] is said to be a key polynomial for ν if for every

f ∈ K[x], if ǫ(f ) ≥ ǫ(Q), then deg(f ) ≥ deg(Q). For a polynomial f ∈ K[x] let f = f 0 + f 1 Q + . . . + f r Q r be the Q-expansion of f . We set S Q (f ) = {i, 0 ≤ i ≤ r | ν Q (f ) = ν(f i Q i )} and δ Q (f ) = max S Q (f ).
Throughout this paper we will fix a limit key polynomial F for Ψ n and denote

d := deg X (F ). Set B = lim Q∈Ψn ν(Q) and B = lim Q∈Ψn ν Q (F ). Take Q 0 ∈ Ψ n and choose Q ∈ Ψ n such that (2) ǫ(Q) -ǫ(Q 0 ) > d(B -ν(Q))
and ( 3)

ǫ(Q) -ǫ(Q 0 ) > B -ν Q (F ). Write F = L(Q) for L(X) ∈ K[x] n [X].
The next result is well-known. We will reprove it here because we need this slightly stronger statement. Lemma 2.1. Let Q be a key polynomial and take f such that f = qQ + r with γ = max{ǫ(f ), ǫ(r)} < ǫ(Q). Then we have

ν Q (qQ) -(ǫ(Q) -γ) ≥ ν(f ) = ν(r). Proof. Take b ∈ I(qQ). Since ǫ(qQ) = max{ǫ(Q), ǫ(q)} ≥ ǫ(Q) (Corollary 4.4 of [1]) we have ν(qQ) -bǫ(Q) ≥ ν(∂ b (qQ)) ≥ min{ν(∂ b (f )), ν(∂ b (r))} ≥ min{ν(f ), ν(r)} -bγ. Consequently, ν(f ) = ν(r) and ν(qQ) -ν(f ) ≥ b(ǫ(Q) -γ) ≥ ǫ(Q) -γ.
Applying the above discussion to ν Q instead of ν we obtain the result.

Take f ∈ K[x] and Q ′ ∈ Ψ n with ǫ(Q) ≤ ǫ(Q ′ ) with ǫ(f ) < ǫ(Q 0 ) and write f = qQ ′ + r with deg(r) < deg(Q ′ ) = deg(Q 0 ). By (2), (3), Lemma 2.1 and the fact that ǫ(r) < ǫ(Q 0 ) we have (4) ν Q ′ (qQ ′ ) > ν(f ) + d(B -ν(Q))
and ( 5)

ν Q ′ (qQ ′ ) > ν(f ) + B -ν Q (F ).
The next result is a well-known result about key polynomials.

Lemma 2.2. Take Q, Q ′ ∈ Ψ n be such that ν(Q) < ν(Q ′ ). For f ∈ K[x] let f = r i=0 f i Q ′i be the Q ′ -expansion of f . Then ν Q (f ) = min 0≤i≤r {ν Q (f i Q ′i )}.
Proof. Since Q ′ is monic and has the smallest degree among all polynomials f such that ν Q (f ) < ν(f ), it is a (Mac Lane-Vaquié) key polynomial for ν Q (Theorem 31 of [START_REF] Decaup | Abstract key polynomials and comparison theorems with the key polynomials of Mac Lane-Vaquié[END_REF]). In particular, Q ′ is ν Q -minimal and the result follows from Proposition 2.3 of [START_REF] Nart | Key polynomials over valued fields[END_REF].

Lemma 2.3. Let Q ′ ∈ Ψ n such that ǫ(Q) < ǫ(Q ′ ). For any f ∈ K[x] let f = a 0 + a 1 Q + . . . + a r Q r and f = b 0 + b 1 Q ′ + . . . + b r Q ′r be the Q and Q ′ -expansions of f , respectively. For l = δ Q (f ) we have ν(a l -b l ) > ν(a l ).
In particular, ν(a l ) = ν(b l ).

Proof. Let h := Q -Q ′ so that Q = Q ′ + h. Then a i Q i = i j=0 i j a i h j Q ′i-j .
For each i, 0 ≤ i ≤ r, and j, 0

≤ j ≤ i, let (6) i j a i h j = a ij0 + a ij1 Q ′ + . . . + a ijn Q ′n be the Q ′ expansion of a i h j . Then b l = i-j+k=l a ijk .
For i, 0 ≤ i ≤ r, and j, 0 ≤ j ≤ i, if k := l + j -i > 0, then by (4), we have

ν(a ijk ) + kν(Q ′ ) > ν(a i h j ) + d(B -ν(Q)) ≥ ν(a i Q i ) + (j -i)ν(Q) + k(B -ν(Q)) ≥ ν(a l Q l ) + (j -i)ν(Q) + k(B -ν(Q)) ≥ ν(a l ) + (l + j -i)ν(Q) + k(B -ν(Q)) = ν(a l ) + kB. Since B > ν(Q ′ ) we have ν(a ijk ) > ν(a l ).
Suppose now that k := l+j-i = 0 (i.e., that i = l+j). If j = 0, then by definition

a ijk = a l . If j > 0, then i > l. Since l = δ Q (f ) we have ν(a i Q i ) > ν(a l Q l ).
Then by Lemma 2.2, applied to (6), we have

ν Q (a ijk Q ′k ) ≥ ν Q (a i h j ) = ν(a i Q i ) + (j -i)ν(Q) > ν(a l ) + (l + j -i)ν(Q). Since ν Q (Q ′ ) = ν(Q)
we obtain that ν(a ijk ) > ν(a l ) and the result follows.

For each ρ < λ and f

∈ K[x], let f = a ρ0 (f ) + a ρ1 (f )Q ρ + . . . + a ρr (f )Q r ρ
be the Q ρ -expansion of f . The value of a ρ0 (f ) will be very important in what follows.

Proposition 2.4.

For f ∈ K[x] with deg X (f ) ≤ d, write f = l(Q) for l(X) ∈ K[x] n [X].
For every ρ < λ we have

(7) ν ρ (l(h ρ )) ≥ ν ρ (f ).
Moreover, the equality holds in (7) if and only if

ν (a ρ0 (f )) = ν ρ (f ) = ν(l(h ρ )).
Proof. By definition

f = l(Q) = l(Q ρ + h ρ ) = Q ρ p(x) + l(h ρ ) for some p(x) ∈ K[x]. Hence a ρ0 (f ) = a ρ0 (l(h ρ )). Let (8) l(h ρ ) = a ρ0 (f ) + b 1 Q ρ + . . . + b l Q l ρ be the Q ρ -expansion of l(h ρ ). We will show that ν(b i Q i ρ ) > ν ρ (f ) for every i, 1 ≤ i ≤ l,
and this will imply our result.

Let f = a 0 + a 1 Q + . . . + a r Q r be the Q-expansion of f , so that (9) l(h ρ ) = a 0 + a 1 h ρ + . . . + a r h r ρ .

For each j, 1 ≤ j ≤ r, consider the Q ρ -expansion (10) a j h j ρ = a ρ0j + a ρ1j Q ρ + . . . + a ρlj Q l ρ of a j h j ρ . Comparing (8), ( 9) and (10), it is enough to show that ν a ρij Q i ρ > ν (a ρ0 (f )) for every i, j, 1 ≤ i ≤ l and 1 ≤ j ≤ r.

For a fixed j, 1 ≤ j ≤ r, by (4) applied to (10) we have

(11) ν a ρij Q j ρ > ν a j Q j + s(B -ν(Q)). Since ν(Q) = ν(h ρ ) = ν ρ (Q), if ν(a 0 ) < ν(a i Q i ) for every i, 1 ≤ i ≤ r, then ν ρ (f ) = ν(a 0 ) = ν(l(h ρ ))
and we are done. Suppose not and take l = δ Q (f ) > 0. By (11) and the fact that ν(a ρl (f )) = ν(a l ) (Lemma 2.3), we have

ν a ρij Q j ρ > ν(a j Q j ) + l(B -ν(Q)) ≥ ν(a l ) + lν(Q) + l(B -ν(Q)) = ν(a l ) + lB > ν(a l ) + lν(Q ρ ) ≥ ν ρ (f ).
This completes the proof.

ν(l(h σ )) = ν(f ) = ν σ (f ) = ν(a σ0 (f ))
for every σ, ρ < σ < λ.

Proof. It is well-known that if f is Ψ n -stable, then there exists Q ′ ∈ Ψ n such that 0 ∈ S Q ′ (f ). The result follows immediately.

The Taylor expansion of a polynomial

We will consider the ring K(x)[X] where X is an indeterminate and let ∂ i denote the i-th Hasse derivative with respect to X. Then, for every l(X) ∈ K(x)[X] and a, b ∈ K[x] we have the Taylor expansion [START_REF] Kaplansky | Maximal fields with valuations I[END_REF]). Let Γ be an ordered abelian group, β 1 , . . . , β n ∈ Γ and {γ ρ } ρ<λ an increasing sequence in Γ, without a last element. If t 1 , . . . , t n are distinct positive integers, then there exist b, 1 ≤ b ≤ n, and ρ < λ, such that

l(b) = l(a) + deg X l i=1 ∂ i l(a)(b -a) i . Lemma 3.1 (Lemma 4 of
β i + t i γ σ > β b + t b γ σ for every i, 1 ≤ i ≤ n, i = b and σ > ρ. Proposition 3.2. Take f ∈ K[x] such that deg X (f ) ≤ d and write f = l(Q) for l(X) ∈ K[x] n [X].
Then there exists ρ < λ such that ν σ (f ) = ν(l(h σ )) for every σ, ρ < σ < λ.

In particular, f is Ψ n -stable if and only if it is (Ψ n , Q)-fixed.

Proof. Since for every j, 1 ≤ j ≤ deg X (l), we have deg(∂ j l(Q)) < deg(F ) we can use Corollary 2.5 to obtain ρ < λ such that (12) β j := ν σ (∂ j l(h σ )) = ν (∂ j l(h σ )) for every j, 1 ≤ j ≤ deg X (l) and ρ < σ. By Lemma 3.1, there exist b, 1 ≤ b ≤ deg X (l), and ρ < λ such that for every λ, ρ < σ < λ and i = b we have ( 13)

β b + bγ ρ < β i + iγ ρ and (12) is satisfied. For σ, ρ < σ < λ, since f -l(h σ ) = deg X (f ) i=1 ∂ i l(h σ )Q i σ we have (14) ν σ (f -l(h σ )) = β b + bγ σ and (15) ν (f -l(h σ )) = β b + bγ σ .
By Proposition 2.4 we have ν σ (l(h σ )) ≥ ν σ (f ). This and (14) imply that ( 16)

ν(l(h σ )) ≥ ν σ (l(h σ )) ≥ β b + bγ σ . If ν σ (f ) = ν(f ), then ν(l(h σ )) ≥ ν(f ).
Suppose, aiming for a contradiction, that ν(l(h σ )) > ν(f ). Then by (15) we have

ν(f ) = β b + bγ σ .
For any σ ′ > σ, by ( 16) we would obtain

ν(l(h σ ′ )) ≥ β b + bγ σ ′ > β b + bγ σ = ν(f )
and this contradicts (15) (with σ replaced by σ ′ ). Hence, ν(l

(h σ )) = ν(f ). Suppose now that ν σ (f ) < ν(f ). If ν σ (l(h σ )) > ν σ (f ), then by (14) we have ν σ (f ) = β b + bγ σ . Consequently, ν(f -l(h σ )) ≥ min{ν(f ), ν(l(h σ ))} > ν σ (f ) = β b + bγ σ ,
what is a contradiction to (15). Hence,

ν σ (l(h σ )) = ν σ (f ).
By the second part of Proposition 2.4 we obtain that ν σ (f ) = ν(l(h σ )). This, ( 14)-( 16) and the fact that ν σ (f ) < ν(f ) imply that

ν σ (f ) = ν(l(h σ )) = β b + bγ σ .
Remark 3.3. In the proof of Theorem 1.1 we will use the explicit calculation of ν(l(h σ )) obtained in the previous proposition.

Proof of Theorem 1.1

We will adapt the proof by Kaplansky in [START_REF] Kaplansky | Maximal fields with valuations I[END_REF]. For each i, 1 ≤ i ≤ d, the polynomial ∂ i L(Q) has degree smaller than deg(F ), hence by Corollary 2.5 there exists ρ 0 < λ such that (17)

β i := ν(∂ i L(Q)) = ν(∂ i L(h ρ ))
for every ρ, ρ 0 < ρ < λ.

Lemma 4.1. If i = p t and j = p t r with r > 1 and p ∤ r, then there exists ρ < λ such that

β i + iγ σ < β j + jγ σ for every σ, ρ < σ < λ.
Moreover, if C in the value group of ν is such that C > γ ρ for every ρ < λ, then

β i + iC < β j + jC.
Proof. From the Taylor formula (applied to ∂ i L) we have

∂ i L(h σ ) -∂ i L(h ρ ) = n-i k=1 ∂ k ∂ i L(h ρ )(h σ -h ρ ) k = n-i k=1 i + k i ∂ i+k L(h ρ )(h σ -h ρ ) k .
By Lemma 3.1, for ρ < σ large enough

(18) ν (∂ i L(h σ ) -∂ i L(h ρ )) = min 1≤k≤n-i ν i + k i ∂ i+k L(h ρ )(h σ -h ρ ) k .
In particular, taking k = j -i, this gives

(19) ν (∂ i L(h σ ) -∂ i L(h ρ )) ≤ ν j i ∂ j L(h ρ )(h σ -h ρ ) j-i
By ( 17) and ( 19) we have

β i ≤ ν(∂ i L(h σ ) -∂ i L(h ρ )) ≤ ν j i ∂ j L(h ρ )(h σ -h ρ ) j-i = ν j i + β j + (j -i)γ ρ .
Since p ∤ j i and char(Kν) = p we have ν j i = 0. Consequently,

β i ≤ β j + (j -i)γ ρ .
This means that for every σ, ρ < σ < λ, we have

β i + iγ σ < β j + jγ σ .
Take C > γ ρ for every ρ < λ. If β i + iC ≥ β j + jC, then

β i -β j ≥ (j -i)C > (j -i)γ σ for every ρ < λ
and this contradicts the first part.

The proof of the next result is very similar to the proof of Proposition 2.4.

Lemma 4.2. Fix θ < λ and for each i,

0 ≤ i ≤ r, set a i0 := a θ0 (∂ i L(h θ )). Then ν θ (∂ i L(h θ ) -a i0 ) + iν(Q θ ) > B. Proof. Since F = a 0 + a 1 Q + . . . + a r Q r we have ∂ 0 L(h θ ) = a 0 + a 1 h θ + . . . + a r h r θ ∂ 1 L(h θ ) = a 1 + . . . + ra r h r-1 θ . . . ∂ r L(h θ ) = a r . If we write ∂ i L(h θ ) = b i0 + b i1 h θ + . . . + b is h s θ , then ν b ij h j θ + iν(Q) ≥ ν(a i+j Q i+j ) ≥ ν Q (F ). For each j > 1, write b ij h j θ = a i0j + a i1j Q θ + . . . + a isj Q s θ .
For every i, j and k > 0, by [START_REF] Moraes | Limit key polynomials as p-polynomials[END_REF] we have

ν a ikj Q k θ + iν(Q θ ) > ν(b ij h j θ ) + B -ν Q (F ) + iν(Q θ ) > ν Q (F ) + B -ν Q (F ) = B.
For every i, we have

∂ i L(h θ ) -a i0 = k,j>0 a ikj Q j θ .
The result follows.

We proceed now with the proof of Theorem 1.1.

Proof of Theorem 1.1. For each i = p s with 1 ≤ i ≤ deg(F ) and j = p s r, with p ∤ r, by Lemma 4.1 we have (20) β i + iB < β j + jB for ρ large enough.

Then there exists ρ ij such that for every σ > ρ ij we have (21) β i + iγ ρ < β j + jγ σ for every ρ < λ.

Take σ such that (21) is satisfied for every i = p s and j = p s r, with p ∤ r. Write I = {l | 1 ≤ l ≤ d such that l = p i for some i ∈ N} and J := {1, . . . , d} \ I.

Then, for every j ∈ J there exists i ∈ I such that β i + iγ ρ < β j + jγ σ for every ρ < λ.

This means that for every ρ > σ we have In order to prove (i), take θ > σ and consider the polynomial

F p (x) = L(h θ ) + i∈I ∂ i L(h θ )(Q -h θ ) i =: L p (Q). Then L(h ρ ) -L p (h ρ ) = j∈J ∂ j L(h θ )(h ρ -h θ ) j .
Consequently, ν (L(h ρ ) -L p (h ρ )) > β b + bγ ρ = ν(L(h ρ )).

The last equality follows from the proof of Proposition 3.2 (as observed in Remark 3.3). Hence, ν(L p (h ρ )) = β b + bγ ρ .

If r / ∈ I, then deg X (F p ) < d so we apply Lemma 3.2 to obtain that F p ∈ S n . This is a contradiction to the minimality of the degree of F in S n . Hence, deg X (F p ) = d and F p is monic. Since deg X (F p ) ≤ d, by Lemma 3.2 we obtain that F p ∈ S n and so is a limit key polynomial for Ψ n .

In order to prove (ii), for each i ≥ 0 take

a i = a θ0 (∂ i L(h θ ))
and

F p := i∈I∪{0} a i Q i θ .
By Lemma 4.2 we have ρ F p -F p > B > ν ρ (F ) for every ρ, θ < ρ < λ.

As before, we conclude that F p is a limit key polynomial for Ψ n and this completes the proof.

Remark 4.3. One can prove (Proposition 3.5 of [START_REF] Moraes | Limit key polynomials as p-polynomials[END_REF]) that a r = 1 and in particular deg(F ) = n deg X (F ).

Corollary 2 . 5 .

 25 If deg(f ) < deg(F ), then there exists ρ such that

∂

  j L(h σ )(h ρ -h σ ) j   ≥ min j∈J {β j + jγ σ } > min i∈I {β i + iγ ρ } = β b + iγ b .

During the realization of this project the first author was supported by a grant from Fundação de Amparo à Pesquisa do Estado de São Paulo (process number 2017/17835-9).


Silva De Souza

J A And Spivakovsky

GRADED RINGS ASSOCIATED TO VALUATIONS AND DIRECT LIMITS

In this paper, we study the structure of the graded ring associated to a limit key polynomial Qn in terms of the key polynomials that define Qn. In order to do that, we use direct limits. In general, we describe the direct limit of a family of graded rings associated to a totally ordered set of valuations. As an example, we describe the graded ring associated to a valuation-algebraic valuation as a direct limit of graded rings associated to residue-transcendental valuations.

Introduction

The graded ring structure associated to a valuation ν, denoted by G ν (see Definition 3.1), has proved to be an important object on Valuation Theory. For example, the graded ring describes information of the value group νK and the residue field Kν simultaneously. It was proved in [START_REF] Barnabé | On the structure of the graded algebra associated to a valuation[END_REF] that G ν is isomorphic to the semigroup ring Kν[t νK ] with a suitable multiplication. Also, this structure is related to the program developed by Teissier to prove local uniformization, an open problem in positive characteristic with applications in resolution of singularities. This program is based on the study of the spectrum of certain graded rings (see [START_REF] Teissier | Valuations, deformations and toric geometry[END_REF]).

Other important objects, which are also linked with programs to prove local uniformization, are key polynomials (see Definition 2.5). These polynomials were introduced by Mac Lane in [START_REF] Maclane | A Construction for Absolute Values in Polynomial Rings[END_REF] and generalized years later by Vaquié in [START_REF] Vaquié | Extension d'Une Valuation[END_REF], using the structure of graded ring. We will refer to them as Mac Lane-Vaquié key polynomials. After that, Novacoski and Spivakovsky in [START_REF] Novacoski | Key polynomials and pseudo-convergent sequences[END_REF] and Decaup, Mahboub and Spivakovsky in [START_REF] Decaup | Abstract key polynomials and comparison theorems with the key polynomials of Mac Lane-Vaquié[END_REF] introduced a new notion of key polynomials, which is the one we use in this paper. These two definitions can be well understood by using graded rings, as one can see in [START_REF] Bengus-Lasnier | Minimal Pairs, Truncation and Diskoids[END_REF] and [START_REF] Novacoski | On MacLane-Vaquié key polynomials[END_REF].

Among key polynomials, the so called limit key polynomials are of great interest to us. Limit key polynomials were introduced in [START_REF] Vaquié | Extension d'Une Valuation[END_REF] and are one of the main aspects of the generalization of Mac Lane's original key polynomials by Vaquié. Here we use a formulation similar to the one presented in [START_REF] Novacoski | Key polynomials and pseudo-convergent sequences[END_REF] (see Definition 5.2).

2010 Mathematics Subject Classification. Primary 13A18. During the realization of this project the authors were supported by a grant from Fundação de Amparo à Pesquisa do Estado de São Paulo (process numbers 2017/17835-9 and 2021/13531-0).

These polynomials are related to the existence of defect, which is an obstacle when dealing with valuations and local uniformization. For example, in the case where the valuation has a unique extension, the defect is the product of the relative degrees of limit key polynomials (see [START_REF] Nart | MacLane-Vaquié chains of valuations on a polynomial ring[END_REF], [START_REF] Saturnino | Defect of an extension, key polynomials and local uniformization[END_REF] or [START_REF] Vaquié | Famille admissible de valuations et defaut d'une extension[END_REF]).

For a valuation ν on K[x], we consider the set Ψ n of all key polynomials for ν of degree n. In this paper, we study the structure of the graded ring associated to a limit key polynomial Q n for Ψ n , denoted by G Qn , in terms of the key polynomials Q ∈ Ψ n . A related problem was studied in [START_REF] Vaquié | Extension d'Une Valuation[END_REF]. Here, we approach this problem by describing G Qn as the direct limit of a direct system defined by the graded rings G Q and the maps presented in Section 3.

Take a valuation ν 0 on K with value group Γ 0 . Fix a totally ordered divisible group Γ containing Γ 0 . Take Γ ∞ := Γ ∪ {∞} with the usual extension of addition and order. Let

V = {ν 0 } ∪ {ν : K[x] → Γ ∞ | ν is a valuation extending ν 0 }.
Consider the partial order on V given by ν 0 ≤ ν for every ν ∈ V and, for ν, µ ∈

V \ {ν 0 }, we set ν ≤ µ if and only if ν(f ) ≤ µ(f ) for every f ∈ K[x].
Our first result deals with an arbitrary totally ordered subset v = {ν i } i∈I ⊂ V such that there exists ν ∈ V satisfying ν i ≤ ν for every i ∈ I. Theorem 4.7 will give us that lim -→ G νi is isomorphic to the additive subgroup R of G ν generated by the set

{in ν (f ) | f is v-stable} (v-stability is defined in Section 4).
Next, we divide the totally ordered subsets v = {ν i } i∈I ⊂ V into three types: the ones with maximum, the ones without maximum such that every f ∈ K[x] is vstable and the ones without maximum such that there exists at least one polynomial that is not v-stable. We show that in the first and second cases there exists ν ∈ V, that we will denote by sup i∈I ν i , satisfying ν ≥ ν i for every i ∈ I and R = G ν (Proposition 4.9 and Corollary 4.11). In the third case, we show that for a polynomial Q of smallest degree that is not v-stable we can define µ ∈ V such that µ is equal to its truncation at

Q (see Definition 2.7), µ ≥ ν i for every i ∈ I and R = R Q , where R Q = {in µ (f ) | deg(f ) < deg(Q)} ⊂ G µ (Proposition 4.
12 and Corollary 4.13).

We then give two applications of the previous results. The first one concerns limit key polynomials, our main interest. We prove that, given a limit key polynomial Q n for Ψ n , the subset v = {ν Q } Q∈Ψn ⊂ V is totally ordered without a maximum and Q n is a polynomial of smallest degree that is not v-stable (Corollary 5.5). Therefore, lim

-→ G νi ∼ = R Qn (Corollary 5.6).
The second application concerns valuation-algebraic valuations (see Definition 6.2). We prove that, given a valuation-algebraic valuation ν, there exists a totally ordered subset v = {ν Q } Q∈Q ⊂ V without maximum with each Q a key polynomial for ν, ν Q a residue-transcendental valuation and ν = sup Q∈Q ν Q (Proposition 6.9).

Therefore, lim

-→ G Q ∼ = G ν (Corollary 6.10).
This paper is organized as follows. In Section 2, we present the main definitions and results that will be used throughout the paper. In Section 3, we present the main results about graded rings associated to a valuation that will be useful in our discussions. In Section 4, for a given totally ordered subset v = {ν i } i∈I ⊂ V, we begin presenting some properties of the direct limit of the direct system {(G νi , φ ij )} i,j∈I i≤j and prove Theorem 4.7. Then we prove Corollary 4.11 and Corollary 4.13 in Subsections 4.1 and 4.2, respectively. In Section 5, we describe the graded ring associated to a limit key polynomial via Corollary 5.6. In Section 6, we describe the graded ring associated to a valuation-algebraic valuation via Corollary 6.10.

Acknowledgements. We would like to thank the anonymous referee for carefully reading, providing useful suggestions and pointing out a few mistakes in an earlier version of this paper.

Preliminaries

Definition 2.1. Take a commutative ring R with unity. A valuation on R is a mapping ν : R -→ Γ ∞ := Γ ∪ {∞} where Γ is a totally ordered abelian group (and the extension of addition and order to ∞ is done in the natural way), with the following properties: Remark 2.2. Take a valuation ν on a field K and a valuation ν on K, the algebraic closure of K, such that ν| K = ν. Then νK is a divisible group. Additionally, νK = νK ⊗ Z Q (see [START_REF] Engler | Valued Fields[END_REF], p.79). About the residue fields, it is known that Kν is the algebraic closure of Kν (see [START_REF] Engler | Valued Fields[END_REF], p.66).

Fix a valuation ν on K[x], the ring of polynomials in one indeterminate over the field K. Our main definition of key polynomial relates to the one in [START_REF] Novacoski | Key polynomials and pseudo-convergent sequences[END_REF], which is related to the one in [START_REF] Herrera Govantes | Key polynomials for simple extensions of valued fields[END_REF]. Fix an algebraic closure K for K and fix a valuation ν on

K[x] such that ν| K[x] = ν. Definition 2.3. Let f ∈ K[x] be a non-zero polynomial. • If deg(f ) > 0, set δ(f ) := max{ν(x -a) | a ∈ K and f (a) = 0}. • If deg(f ) = 0, set δ(f ) = -∞.
Remark 2.4. According to [START_REF] Novacoski | Key polynomials and minimal pairs[END_REF], δ(f ) does not depend on the choice of the algebraic closure K or the extension ν of ν.

Definition 2.5. A monic polynomial Q ∈ K[x] is a key polynomial of level δ(Q) if, for every f ∈ K[x], δ(f ) ≥ δ(Q) =⇒ deg(f ) ≥ deg(Q).
Let q ∈ K[x] be a non-constant polynomial. Then there exist uniquely determined polynomials f 0 , . . . ,

f s ∈ K[x] with deg(f i ) < deg(q) for every i, 0 ≤ i ≤ s, such that (1) f = f 0 + f 1 q + . . . + f s q s .
We call this expression the q -expansion of f . Proposition 2.6.

(Lemma 2.3 (iii) of [13] + Corollary 3.52 of [7]) Let Q ∈ K[x]
be a monic polynomial. The following assertions are equivalent.

(i): Q is a key polynomial for ν. (ii): For every f, g ∈ K[x] with deg(f ) < deg(Q) and deg(g) < deg(Q), if f g = lQ + r is the Q-expansion of f g, then ν(f g) = ν(r) < ν(lQ).
Definition 2.7. Let q ∈ K[x] be a non-constant polynomial and ν a valuation on

K[x]. The map ν q (f ) := min 0≤i≤s {ν(f i q i )}, for f ∈ K[x] as in (1), is called the truncation of ν at q.
This map is not always a valuation, as we can see in Example 2.4 of [START_REF] Novacoski | Key polynomials and pseudo-convergent sequences[END_REF].

Proposition 2.8. (Proposition 2.6 of [10]) If Q is a key polynomial, then ν Q is a valuation on K[x].
In the next lemmas, we state some properties of key polynomials and truncations. In what follows, we denote by Γ

Q := ν(K[x]) ⊗ Z Q the divisible hull of ν(K[x]). Lemma 2.9. (Proposition 2.10 of [13]) Let Q, Q ∈ K[x]
be key polynomials for ν. We have the following.

(1) If deg(Q) < deg(Q ), then δ(Q) < δ(Q ). (2) If δ(Q) < δ(Q ), then ν Q (Q ) < ν(Q ). (3) If deg(Q) = deg(Q ), then ν(Q) < ν(Q ) ⇐⇒ ν Q (Q ) < ν(Q ) ⇐⇒ δ(Q) < δ(Q ).
Lemma 2.10. (Corollaries 3.9, 3.10, 3.11 and 3.13 of [START_REF] Novacoski | On MacLane-Vaquié key polynomials[END_REF])

Let Q, Q ∈ K[x] be key polynomials such that δ(Q) ≤ δ(Q ). (1) We have ν Q (Q) = ν(Q). (2) For every f ∈ K[x], we have ν Q (f ) ≤ ν Q (f ). In particular, if δ(Q) = δ(Q ) then ν Q = ν Q . (3) For every f ∈ K[x], if ν Q (f ) = ν(f ), then ν Q (f ) = ν(f ). (4) For every f ∈ K[x], if δ(Q) < δ(Q ) and ν Q (f ) < ν(f ), then ν Q (f ) < ν Q (f ).
Remark 2.11. In [START_REF] Novacoski | Key polynomials and pseudo-convergent sequences[END_REF] and [START_REF] Novacoski | On MacLane-Vaquié key polynomials[END_REF] the definition of key polynomial and the above results are stated using the notion of (f ) instead of δ(f ). For a non-zero polynomial

f ∈ supp(ν) with deg(f ) > 0, one defines (f ) := max 1≤b≤deg(f ) ν(f ) -ν(∂ b f ) b ∂ b f ∈ supp(ν) ∈ Γ Q ,
where ∂ b f is the formal Hasse-derivative of order b of f . If f ∈ supp(ν) and deg(f ) = 0, then we set (f ) := -∞ and if f ∈ supp(ν), then we set (f ) := ∞. Proposition 3.1 of [START_REF] Novacoski | Key polynomials and minimal pairs[END_REF] shows that δ(f

) = (f ) for all f ∈ K[x].

Graded ring associated to a valuation

Let ν be a valuation on K

[x]. For each γ ∈ ν(K[x]
), we consider the abelian groups

P γ = {f ∈ K[x] | ν(f ) ≥ γ} and P + γ = {f ∈ K[x] | ν(f ) > γ}.
Definition 3.1. The graded ring associated to ν is defined by

G ν = gr ν (K[x]) := γ∈ν(K[x]) P γ /P + γ .
The sum on G ν is defined coordinatewise and the product is given by extending the product of homogeneous elements, which is described by

f + P + β • g + P + γ := f g + P + β+γ ,
where β = ν(f ) and γ = ν(g).

For f ∈ supp(ν), we denote by in ν (f ) the image of f in P ν(f

) /P + ν(f ) ⊆ G ν . If f ∈ supp(ν)
, then we define in ν (f ) = 0. The next lemma follows from the definitions above.

Lemma 3.2. Let f, g ∈ K[x].
We have the following.

(1) G ν is an integral domain. [START_REF] Barnabé | Valuations on K[x] approaching a fixed irreducible polynomial[END_REF]

in ν (f ) • in ν (g) = in ν (f g). (3) in ν (f ) = in ν (g) if and only if ν(f -g) > ν(f ) = ν(g).
Let ν i and ν j be valuations on K

[x] such that ν i (f ) ≤ ν j (f ) for all f ∈ K[x]. Let P γ (K[x], ν i ) = {f ∈ K[x] | ν i (f ) ≥ γ} (analogously we define P γ (K[x], ν j ), P + γ (K[x], ν i ) and P + γ (K[x], ν j )).
We have the inclusions

P γ (K[x], ν i ) ⊆ P γ (K[x], ν j ) and P + γ (K[x], ν i ) ⊆ P + γ (K[x], ν j ) for any γ ∈ ν i (K[x]) ⊆ ν j (K[x]
). We consider the following map:

φ ij : G νi -→ G νj (2) in νi (f ) -→ in νj (f ) if ν i (f ) = ν j (f ) 0 if ν i (f ) < ν j (f ),
and we extend this map naturally for an arbitrary element. This map is well-defined (Corollary 5.5 of [START_REF] Bengus-Lasnier | Minimal Pairs, Truncation and Diskoids[END_REF]) and, by construction, it is a homomorphism of graded rings.

Suppose that q ∈ K[x] is such that ν q is a valuation. Let R q be the additive subgroup of G νq generated by the set

{in νq (f ) | f ∈ K[x] d }, where d = deg(q) and K[x] d = {f ∈ K[x] | deg(f ) < d}.
We set y q := in νq (q). The next propositions say that a non-zero y q can be seen as a transcendental element over R q and that R q is a subring of G νq if and only if q is a key polynomial for ν.

G νq = R q [y q ].
Moreover, if q ∈ supp(ν), then G νq = R q . If q ∈ supp(ν), then y q is such that if a 0 + a 1 y q + . . . + a s y s q = 0 for some a 0 , . . . , a s ∈ R q , then a i = 0 for every i, 0 ≤ i ≤ s. Proposition 3.4. (Theorem 5.7 of [START_REF] Novacoski | On truncations of valuations[END_REF]) Suppose ν q is a valuation on K[x]. Then the following assertions are equivalent.

(i): q is a key polynomial for ν. (ii): The set R q is a subring of G νq .

Totally ordered sets of valuations and direct limits

Take V as in the introduction, that is,

V = {ν 0 } ∪ {ν : K[x] → Γ ∞ | ν is a valuation extending ν 0 }.
Consider the partial order on V given by ν 0 ≤ ν for every ν ∈ V and, for ν, µ ∈ V \ {ν 0 }, we set ν ≤ µ if and only if ν(f ) ≤ µ(f ) for every f ∈ K[x]. We say that ν < µ if ν ≤ µ and there exists g ∈ K[x] such that ν(g) < µ(g). In V we have the following property.

Proposition 4.1. (Proposition 2.2 of [2]) Assume η, ν, µ ∈ V are such that η < ν < µ. For f ∈ K[x], if η(f ) = ν(f ), then ν(f ) = µ(f ).
Let v = {ν i } i∈I ⊂ V be a totally ordered set. Consider a total order on the index set I induced from the order on v. Since we have a total order, (I, ≤) is a directed set. 1 Lemma 4.2. Let v = {ν i } i∈I be a totally ordered set in V. Consider the family of graded rings {G νi } i∈I . For ν i ≤ ν j , let φ ij be the map

φ ij : G νi -→ G νj in νi (f ) -→ in νj (f ) if ν i (f ) = ν j (f ) 0 if ν i (f ) < ν j (f ),
extended in a natural way to arbitrary (that is, not necessarily homogeneous) elements of G νi . Then {(G νi , φ ij )} i,j∈I i≤j is a direct system over I. Proof. We need to check that φ ii is the identity map and

φ ik = φ jk • φ ij for all i ≤ j ≤ k. • By definition, φ ii (in νi (f )) = in νi (f ) for every f ∈ K[x], hence φ ii is the identity map on G νi . • Take i ≤ j ≤ k, that is, (3) ν i (f ) ≤ ν j (f ) ≤ ν k (f ) for all f ∈ K[x].
If the strict inequality holds in some of the inequalities of (3), then

(φ jk • φ ij )(in νi (f )) = 0 = φ ik (in νi (f )). If ν i (f ) = ν j (f ) = ν k (f ), then (φ jk • φ ij )(in νi (f )) = in ν k (f ) = φ ik (in νi (f )).
Hence,

φ ik = φ jk • φ ij for all i ≤ j ≤ k.
Therefore, {(G νi , φ ij )} i,j∈I i≤j is a direct system over I.

1 That is, ≤ is reflexive and transitive relation on v such that, for every ν i , ν j ∈ v, there exists

ν k ∈ v satisfying ν i ≤ ν k and ν j ≤ ν k .
Remark 4.3. We do not gain in generality if we suppose v = {ν i } i∈I simply a directed set, because every directed set in V is totally ordered. Indeed, if v ⊂ V is a directed set, then given ν, µ ∈ v there exists η ∈ v such that ν ≤ η and µ ≤ η. By Theorem 2.4 of [START_REF] Nart | MacLane-Vaquié chains of valuations on a polynomial ring[END_REF], the set (-∞, η) = {ρ ∈ V | ρ < η} is totally ordered. Therefore, ν and µ are comparable.

We want to describe the direct limit of the direct system {(G νi , φ ij )} i,j∈I i≤j . We are going to use the characterization of lim -→ G νi as a disjoint union. The direct limit of the direct system {(G νi , φ ij )} i,j∈I i≤j is defined as

lim -→ G νi := i∈I G νi / ∼ ,
where ∼ is the following equivalence relation: for a i ∈ G νi and a j ∈ G νj with i ≤ j,

a i ∼ a j ⇐⇒ φ ij (a i ) = a j .
We denote by [a i ] the equivalence class of a i in lim -→

G νi . The operations on lim

-→
G νi are induced from each G νi . Denoting by 0 i the additive identity of G νi , it is easy to see that [0 i ] = [0 j ] for all i, j ∈ I. We write only [0] to denote [0 i ], which is the additive identity of lim [START_REF] Barnabé | On the structure of the graded algebra associated to a valuation[END_REF]] for all i, j ∈ I. The equivalence class [in νi [START_REF] Barnabé | On the structure of the graded algebra associated to a valuation[END_REF]] is the multiplicative identity of lim -→ G νi . We also have the following. (1) We have ν k (f ) < ν j (f ) for some j > k if and only if

-→ G νi . Similarly, if in νi (1) is the multiplicative identity of G νi , it is easy to see that [in νi (1)] = [in νj
[in ν k (f )] = [0]. (2) For j ≥ k, if ν k (f ) = ν j (f ), then [in ν k (f )] = [in νj (f )]. Moreover, if ν k (f ) = ν j (f ) for every j ≥ k, then [in ν k (f )] = [0].
Proof.

(1) If ν k (f ) < ν j (f ) for some j > k, then

φ kj (in ν k (f )) = 0 j . Hence, [in ν k (f )] = [0]. On the other hand, if [in ν k (f )] = [0], then there exists j ≥ k such that φ kj (in ν k (f )) = 0 j .
By the definition of φ kj , this implies ν k (f ) < ν j (f ) and k < j.

(2) For j ≥ k, if ν k (f ) = ν j (f ), then φ kj (in ν k (f )) = in νj (f ). Hence, we have [in ν k (f )] = [in νj (f )]. Moreover, if ν k (f ) = ν j (f ) for every j ≥ k, then by the preceding item we have [in ν k (f )] = [0]. Let v = {ν i } i∈I be a totally ordered subset of V. For every f ∈ K[x], we say that f is v-stable if there exists i f ∈ I such that (4) ν i (f ) = ν i f (f ) for every i ∈ I with i ≥ i f .
In the following, let v = {ν i } i∈I be a totally ordered subset of V and suppose there exists ν ∈ V such that ν i ≤ ν for every i ∈ I. Lemma 4.5. Fix k ∈ I. We have the following.

(1) If ν k (f ) < ν(f ), then ν k (f ) < ν j (f ) for every j > k. (2) We have ν k (f ) = ν(f ) if and only if ν k (f ) = ν j (f ) for every j ≥ k. In other words, f is v-stable if and only if ν k (f ) = ν(f ) for some k ∈ I.
Proof.

(1) If there exists j > k such that ν k (f ) = ν j (f ), then ν k (f ) = ν j (f ) < ν(f ) and ν k < ν j < ν, contradicting Proposition 4.1.

(

) Suppose ν k (f ) = ν(f ). If there exists j > k such that ν k (f ) < ν j (f ), then ν(f ) = ν k (f ) < ν j (f ), which is a contradiction. Conversely, suppose ν k (f ) = ν j (f ) for every j ≥ k. If ν k (f ) < ν(f ), 2
then by the preceding item we would have ν k (f ) < ν j (f ) for every j > k, contradicting our assumption.

Let S ⊂ G ν be any subset. We denote by S the additive subgroup of G ν generated by S. We define

R = {in ν (f ) | f is v-stable} ⊆ G ν .
Lemma 4.6. We have the following.

(1) R is a subring of G ν .

(2) Let d ∈ N ∪ {∞} be the smallest positive integer for which some polynomial of degree is not v-stable (if there is no such polynomial, set d = ∞). Then

R = {in ν (f ) | f ∈ K[x] d } .
Proof.

(1) By construction, R is an additive subgroup of G ν and clearly in ν (1) ∈ R.

For in ν (f ), in ν (g) ∈ R, we take k = max{i f , i g }. Then, for every j ≥ k, it follows that

ν j (f g) = ν j (f ) + ν j (g) = ν k (f ) + ν k (g) = ν k (f g).
That is, f g is v-stable and then in ν (f g) ∈ R. This shows that R is a subring of G ν .

(

) If d = ∞, then K[x] d = K[x] 2
and every f is v-stable. Hence, the result follows. Suppose d < ∞ and take

Q ∈ K[x] not v-stable and deg(Q) = d. We denote R = {in ν (f ) | f ∈ K[x] d } . We have R ⊆ R since deg(f ) < d implies f is v-stable (by the minimality of d = deg(Q)).
Consider in ν (f ) ∈ R, hence f is v-stable. Let f = gQ + f 0 the euclidean division of f by Q. Thus, deg(f 0 ) < d and f 0 is also v-stable. By Lemma 4.5 and the fact that Q is not v-stable, we have that ν(Q) > ν i (Q) for every i ∈ I. Take k = max{i f , i f0 }. Hence, ν k (f ) = ν j (f ) and ν k (f 0 ) = ν j (f 0 ) for every j ≥ k. By Lemma 4.5, ν(f ) = ν k (f ) and ν(f 0 ) = ν k (f 0 ). Thus,

ν(f -f 0 ) = ν(gQ) > ν k (gQ) ≥ min{ν k (f ), ν k (f 0 )} = min{ν(f ), ν(f 0 )}. Hence, ν(f -f 0 ) > ν(f ) = ν(f 0 ), that is, in ν (f ) = in ν (f 0 ) ∈ R . Therefore, R = R .
Now we present our main result. Theorem 4.7. We have lim

-→ G νi ∼ = R.
Proof. Consider the map given by

τ : R -→ lim -→ G νi in ν (f ) -→ [in νi f (f )],
where we extend τ naturally to arbitrary (that is, not necessarily homogeneous) elements of R by additivity. This map is well defined. Indeed, take in ν (f ) ∈ R. By assumption, f is v-stable, so there exists i f ∈ I such that (4) is satisfied. If j f is another index such that ν i (f ) = ν j f (f ) for every i ∈ I with i ≥ j f , then without loss of generality we can take i f ≤ j f . Hence, by Lemma 4.

4 (2), [in νi f (f )] = [in νj f (f )]. Also, suppose f, g ∈ K[x] are such that in ν (f ) = in ν (g) ∈ R. By Lemma 3.2 (3), this means that ν(f -g) > ν(f ) = ν(g). By Lemma 4.6 (2), we can consider f, g ∈ K[x] d . Hence, f -g ∈ K[x] d and f, g and f -g are v-stable. Take k = max{i f , i g , i f -g }. By Lemma 4.5, ν k (f ) = ν(f ), ν k (g) = ν(g) and ν k (f -g) = ν(f -g).
Then,

ν k (f -g) = ν(f -g) > min{ν(f ), ν(g)} = min{ν k (f ), ν k (g)}.
We conclude that in ν k (f ) = in ν k (g). Therefore, using Lemma 4.4 we have

τ (in ν (f )) = [in νi f (f )] = [in ν k (f )] = [in ν k (g)] = [in νi g (g)] = τ (in ν (g))
and we see that τ is well defined.

Since we extended τ to arbitrary elements of R via finite sums, this map is a group homomorphism by construction. We now check that τ is a ring isomorphism.

• τ is injective: since it is a group homomorphism, it is enough to check that ker(τ ) = {0}. Given a non-zero element in ν (f ) ∈ R, we know that

ν i f (f ) = ν j (f ) for every j ≥ i f . By Lemma 4.4 (2), τ (in ν (f )) = [in νi f (f )] = [0].
Hence, ker(τ ) = {0} and τ is injective.

• τ is surjective: take any [in ν k (f )] ∈ lim -→ G νi . If there exists j > k such that ν k (f ) < ν j (f ), then by Lemma 4.4 (1) we have [in νi f (f )] = [0] = τ (0). On the other hand, if ν k (f ) = ν j (f ) for every j ≥ k, then we can take i f = k and [in ν k (f )] = [in νi f (f )] = τ (in ν (f ))
. Therefore, τ is surjective.

• τ is a ring homomorphism: for any in ν (f ), in ν (g) ∈ R, we can take j ∈ I sufficiently large so that j ≥ max{i f g , i f , i g }. We have

τ (in ν (f ) • in ν (g)) = τ (in ν (f g)) = [in νj (f g)] = [in νj (f ) • in νj (g)] = [in νj (f )] • [in νj (g)] = τ (in ν (f )) • τ (in ν (g)).
Also, τ preserves the multiplicative identity since, by definition, τ (in ν (1)) = [in νi [START_REF] Barnabé | On the structure of the graded algebra associated to a valuation[END_REF]] (for any i ∈ I), which is the unity of lim

-→ G νi .
Therefore, we have lim

-→
G νi ∼ = R as commutative rings with unity.

We will classify the totally ordered subsets v ⊂ V in three classes using the following proposition. Proposition 4.8. (Corollary 2.3 of [START_REF] Barnabé | Valuations on K[x] approaching a fixed irreducible polynomial[END_REF]) Let {ν i } i∈I be a totally ordered set in V. For every f ∈ K[x], either {ν i (f )} i∈I is strictly increasing, or there exists i 0 ∈ I such that ν i (f ) = ν i0 (f ) for every i ∈ I with i ≥ i 0 .

We consider three cases:

• v has maximum ν m . • v has no maximum and every f ∈ K[x] is v-stable.
• v has no maximum and there exists q ∈ K[x] not v-stable.

4.1.

First and second cases. Proposition 4.9. Let v = {ν i } i∈I be a totally ordered set in V such that every

f ∈ K[x] is v-stable. Define ν = sup i∈I ν i : K[x] → Γ ∞ by ν(f ) = ν i f (f ). Then ν is a valuation on K[x] such that ν i ≤ ν. Moreover, if ν ∈ V is such that ν ≤ ν and ν i ≤ ν for every i ∈ I, then ν = ν. Proof. Take f, g ∈ K[x]
. Then, by assumption, there exist i f , i g , i f +g and i f g satisfying (4). Take j = max{i f , i g , i f +g , i f g }. Hence,

ν(f g) = ν j (f g) = ν j (f ) + ν j (g) = ν(f ) + ν(g) and ν(f + g) = ν j (f + g) ≥ min{ν j (f ), ν j (g)} = min{ν(f ), ν(g)}. Also, ν(0) = ν i0 (0) = ∞ and ν(1) = ν i1 (1) = 0. Therefore, ν is a valuation on K[x].
In addition, for each f ∈ K[x] and i ∈ I, we have

ν i (f ) ≤ ν i f (f ) = ν(f ). Hence, ν i ≤ ν. Moreover, suppose ν ∈ V is such that ν ≤ ν and ν i ≤ ν for every i ∈ I. Thus, for every f ∈ K[x] we have ν(f ) ≥ ν (f ) ≥ ν i f (f ) = ν(f ). Therefore, ν = ν. Remark 4.10. If v = {ν i } i∈I has maximum ν m , then every f is v-stable (take i f = m).
Hence, ν in Proposition 4.9 coincides with ν m .

We have the following corollary, which covers the first and second cases. 4.2. Third case. Now we treat the third case. Let v = {ν i } i∈I be a totally ordered set in V such that v has no maximum and there is at least one polynomial that is not v-stable. Consider the set

C(v) := {f ∈ K[x] | f is v-stable }.
For every f ∈ C(v) we set v(f ) = ν i f (f ). Let Q be a monic polynomial of smallest degree d not v-stable and take γ ∈ Γ ∞ such that γ > ν i (Q) for every i ∈ I.

Consider the map

µ(f 0 + f 1 Q + . . . + f r Q r ) = min 0≤j≤r {v(f j ) + jγ},
where

f 0 + f 1 Q + . . . + f r Q r is the Q-expansion of f .
Proposition 4.12. We have the following.

(1) We have µ ∈ V.

(2) We have ν i < µ for every i ∈ I.

(3) We have µ = µ Q and Q is a key polynomial for µ.

Proof.

(1) By Theorem 2.4 of [START_REF] Barnabé | Valuations on K[x] approaching a fixed irreducible polynomial[END_REF], µ is valuation. It follows from the definition that ν 0 = µ | K . (2) Using Proposition 1.21 of [START_REF] Vaquié | Extension d'Une Valuation[END_REF] and Theorem 5.1 of [START_REF] Maclane | A Construction for Absolute Values in Polynomial Rings[END_REF], one can prove that

ν i (f ) ≤ µ(f ) for every f ∈ K[x]. Also, ν i (Q) < γ = µ(Q) for every i ∈ I.
Hence, ν i < µ for every i ∈ I. (3) It follows immediately from the definition of µ that µ = µ Q . We now prove that Q is a key polynomial for µ. Take f, g ∈ K[x] with deg(f ) < deg(Q) and deg(g) < deg(Q) and suppose f g = lQ+r is the Q-expansion of f g. We will prove that µ(f g) = µ(r) < µ(lQ). Since deg(f ), deg(g), deg(l), deg(r) < deg(Q), by the minimality of deg(Q) all f, g, l and r are v-stable. Since µ(Q) = γ > ν i (Q) for every i ∈ I, we take k > max{i f , i g , i l , i r } and then

µ(lQ) = µ(l) + µ(Q) > ν k (l) + ν k (Q) ≥ min{ν k (r), ν k (f g)} = min{µ(r), µ(f g)}.
By Proposition 2.6, Q is a key polynomial for µ.

Let R Q be the additive subgroup of G µ generated by the set {in µ (f )

| f ∈ K[x] d }.
Since Q is a key polynomial for µ, Proposition 3.4 guarantees that R Q is a subring of G µ . Corollary 4.13. Let v = {ν i } i∈I be a totally ordered set in V such that v has no maximum and there is at least one polynomial that is not v-stable. Let Q be a monic polynomial of smallest degree d that is not v-stable and take γ ∈ Γ ∞ such that γ > ν i (Q) for every i ∈ I. Take µ as in Proposition 4.12 and R Q as in the above paragraph. Consider the direct system {(G νi , φ ij )} i,j∈I i≤j . Then lim -→

G νi ∼ = R Q as commutative rings with unity.

Proof. By Theorem 4.7 and Lemma 4.6, we have lim

-→ G νi ∼ = R = R Q .
Corollary 4.14. We have

G µ ∼ = (lim -→ G νi )[Y ],
where Y is an indeterminate over lim

-→ G νi . Proof. By Proposition 3.3, G µ = R Q [y Q ], with y Q = in µ (Q). Then, we extend the isomorphism lim -→ G νi ∼ = R Q to an isomorphism (lim -→ G νi )[Y ] ∼ = G µ by sending Y to y Q .

Limit key polynomials and direct limits

Let ν be a valuation on K[x] extending ν 0 . Take n ∈ N and let Ψ n denote the set of all key polynomials for ν with degree n. Assume that Ψ n = ∅ and that {ν(Q) | Q ∈ Ψ n } does not have a maximum. Consider the set

K n := {f ∈ K[x] | ν Q (f ) < ν(f ) for all Q ∈ Ψ n }. Lemma 5.1. Take Q n ∈ K n a monic of polynomial of least degree. Then Q n is a key polynomial for ν. Proof. Consider f 1 , f 2 ∈ K[x] with deg(f 1 ), deg(f 2 ) < deg(Q n ) and suppose f 1 f 2 = f 3 Q n + f 4 is the Q n expansion of f 1 f 2 . Since each f l , 1 ≤ l ≤ 4, has degree smaller than deg(Q n ), it follows by the minimality of deg(Q n ) that f l ∈ K n . Hence, for each l, 1 ≤ l ≤ 4, there exists Q l ∈ Ψ n such that ν Q l (f l ) = ν(f l ). Since {ν(Q) | Q ∈ Ψ n } does not have a maximum, there exists Q ∈ Ψ n such that ν(Q l ) < ν(Q) for all l, 1 ≤ l ≤ 4. By Lemma 2.9 (iii) and Lemma 2.10 (iii), it follows that ν Q (f l ) = ν(f l ) for every l, 1 ≤ l ≤ 4. Thus, ν(f 3 Q n ) = ν(f 3 ) + ν(Q n ) > ν Q (f 3 ) + ν Q (Q n ) = ν Q (f 1 f 2 -f 4 ) ≥ min{ν Q (f 1 f 2 ), ν Q (f 4 )} = min{ν(f 1 f 2 ), ν(f 4 )}. Therefore, ν(f 1 f 2 ) = ν(f 4 ) < ν(f 3 Q n ) and then Q n is a key polynomial for ν by Proposition 2.6. Definition 5.2. A monic polynomial Q n ∈ K[x] is called a limit key polynomial for Ψ n if Q n ∈ K n and Q n has the least degree among polynomials in K n .

Consider the following relation on Ψ

n : (5) Q Q ⇔ ν Q ≤ ν Q and Q ≺ Q ⇔ ν Q < ν Q . We note that if we take Q, Q ∈ Ψ n , then either Q Q or Q Q. Indeed, considering δ(Q), δ(Q ) ∈ Γ Q , since they belong to a totally ordered group, we have δ(Q) ≤ δ(Q ) or δ(Q ) ≤ δ(Q). By Lemma 2.10 (2), for every f ∈ K[x] we have ν Q (f ) ≤ ν Q (f ) or ν Q (f ) ≤ ν Q (f ), that is, Q Q or Q Q.
Therefore, with this pre-order (Ψ n , ) is a directed set. It follows also that v = {ν Q } Q∈Ψn is a totally ordered set. Remark 6.3. By Abhyankar's inequality (see [START_REF] Zariski | Commutative Algebra II[END_REF], p.330), we see that a valuation cannot be value-transcendental and residue-transcendental at the same time. Remark 6.4. Explicitly, a valuation ν on K[x] extending ν 0 is valuation-algebraic if it is a Krull valuation, ν(K[x])/ν 0 K is a torsion group and K(x)ν | Kν 0 is an algebraic field extension. Lemma 6.5. Let ν be a valuation-algebraic valuation on K[x] extending ν 0 on K. Suppose that q is a polynomial such that ν q is a valuation. Then ν q is residuetranscendental.

Proof. By Theorem 3.1 of [START_REF] Novacoski | Key polynomials and minimal pairs[END_REF], we have that ν q is valuation-transcendental. Given f ∈ K[x], f = 0, we know that ν q (f ) = min 0≤i≤r {ν(f i ) + iν(q)} ∈ ν(K[x]), where f 0 , . . . , f r are the coefficients of the q-expansion of f . Since ν is valuationalgebraic, ν(K[x])/ν 0 K is a torsion group. Hence, ν q (f ) is a torsion element in ν q (K[x])/ν 0 K for every f ∈ K[x], f = 0. Therefore, ν q is not value-transcendental and, due to Remark 6.3, we see that ν q is a residue-transcendental valuation.

To a given valuation-algebraic valuation ν, we are going to associate a totally ordered subset of V. In order to do that, we use the results of [START_REF] Novacoski | Key polynomials and pseudo-convergent sequences[END_REF] on complete sets, which we define bellow. Remark 6.8. As remarked in [START_REF] Novacoski | On MacLane-Vaquié key polynomials[END_REF], the definition of complete set in Theorem 1.1 of [START_REF] Novacoski | Key polynomials and pseudo-convergent sequences[END_REF] does not require that deg(Q) ≤ deg(f ). However, the proof of the Theorem shows that this inequality always holds. Proposition 6.9. Let ν ∈ V be a valuation-algebraic valuation. Then there exists a totally ordered subset v = {ν i } i∈I ⊂ V without maximum such that every f ∈ K[x] is v-stable, ν = sup i∈I ν i and each ν i is residue-transcendental.

Proof. By Proposition 6.7, there exists a complete set Q of key polynomials for ν. Consider v = {ν Q } Q∈Q ⊂ V, which is totally ordered due to Lemma 2.10 (2). We order the set Q by posing Q Q if and only if ν Q ≤ ν Q . By Lemma 6.5, each ν Q is residue-transcendental.

We now show that every f ∈ K[x] is v-stable. Indeed, for every f ∈ K[x], there exists Q ∈ Q such that deg(Q) ≤ deg(f ) and ν Q (f ) = ν(f ). If Q Q , then we have the following:

• if δ(Q ) ≤ δ(Q), then ν Q (f ) ≤ ν Q (f ) (Lemma 2.10 (2)), that is, ν Q (f ) = ν Q (f ) = ν(f ); • if δ(Q) < δ(Q ),
(f ) = ν(f ) implies ν Q (f ) = ν(f ). Hence, Q Q implies ν Q (f ) = ν Q (f ) = ν(f ).
That is, f is v-stable and ν = sup Q∈Q ν Q as in Proposition 4.9. Moreover, suppose {ν Q } Q∈Q has a maximum. Then

ν = sup Q∈Q ν Q = ν Qm
for some Q m ∈ Q, which is a contradiction since ν is valuation-algebraic and ν Qm is residue-transcendental. Therefore, {ν Q } Q∈Q does not have a maximum. Corollary 6.10. Let ν ∈ V be a valuation-algebraic valuation and take v = {ν i } i∈I ⊂ V as in Proposition 6.9. Then lim

(

  V1): ν(ab) = ν(a) + ν(b) for all a, b ∈ R. (V2): ν(a + b) ≥ min{ν(a), ν(b)} for all a, b ∈ R. (V3): ν(1) = 0 and ν(0) = ∞. Let ν : R -→ Γ ∞ be a valuation. The set supp(ν) = {a ∈ R | ν(a) = ∞} is called the support of ν. The value group of ν is the subgroup of Γ generated by {ν(a) | a ∈ R \ supp(ν)} and is denoted by νR. A valuation ν is a Krull valuation if supp(ν) = {0}. If ν is a Krull valuation, then R is a domain and we can extend ν to K = Quot(R) on the usual way. In this case, define the valuation ring as O ν := {a ∈ K | ν(a) ≥ 0}. The ring O ν is a local ring with unique maximal ideal m ν := {a ∈ K | ν(a) > 0}. We define the residue field of ν to be the field O ν /m ν and denote it by Kν. The image of a ∈ O ν in Kν is denoted by aν.

Proposition 3 . 3 .

 33 (Proposition 4.5 of[START_REF] Novacoski | On MacLane-Vaquié key polynomials[END_REF]) We have

Lemma 4 . 4 .

 44 For fixed k ∈ I and f ∈ K[x], consider [in ν k (f )] ∈ lim -→ G νi .

Corollary 4 . 11 .

 411 Let v = {ν i } i∈I be a totally ordered set in V such that everyf ∈ K[x] is v-stable. Consider the direct system {(G νi , φ ij )} i,j∈I i≤j . Take ν = sup i∈I ν i as in Proposition 4.9. Then lim -→ G νi ∼ = G ν as commutative rings with unity. Proof. It follows from Theorem 4.7 because R = G ν .

Definition 6 . 6 .

 66 Let ν be a valuation on K[x]. A set Q ⊂ K[x] of key polynomials for ν is called a complete set of key polynomials for ν if for every f ∈ K[x] with deg(f ) ≥ 1 there exists Q ∈ Q with deg(Q) ≤ deg(f ) such that ν Q (f ) = ν(f ).Proposition 6.7. (Theorem 1.1 of[START_REF] Novacoski | Key polynomials and pseudo-convergent sequences[END_REF]) Every valuation ν on K[x] admits a complete set Q of key polynomials.

G

  νi ∼ = G ν as commutative rings with unity. Proof. It follows from Corollary 4.11.

  then by Lemma 2.10 (3) we have that ν Q

Corollary 5.3. Consider the family of graded rings {G Q } Q∈Ψn and, for Q Q , let φ QQ be the map

extended naturally to arbitrary (that is, not necessarily homogeneous) elements of

Proof. This follows from Lemma 4.2 because {ν Q } Q∈Ψn is a totally ordered set.

In the next lemma we gather some properties of limit key polynomials for Ψ n . These properties will allow us to prove that v = {ν Q } Q∈Ψn has no maximum. Lemma 5.4. Let Q n be a limit key polynomial for Ψ n .

(1) We have δ

Proof.

(

. By Lemma 2.9 (1), we would have

(2) By Lemma 2.10, Lemma 2.9 and the preceding item, we have

Corollary 5.5. The family v = {ν Q } Q∈Ψn is a totally ordered subset of V with no maximum. Also, Q n is a polynomial of least degree that is not v-stable. If we take γ = ν(Q n ), then ν Qn (f ) = min 0≤j≤r {v(f j ) + jγ}, where f 0 , . . . , f r are the coefficients of the Q n -expansion of f .

Proof. We already saw that v = {ν Q } Q∈Ψn is a totally ordered set. Suppose, aiming for a contradiction, that v has maximum

. However, by Lemma 5.4 (2), we have ν

and this, together with Lemma 2.10 (4),

Corollary 5.6. We have lim -→

G Q ∼ = R Qn as commutative rings with unity and

Proof. It follows from Corollary 4.13 and Corollary 4.14.

Valuation-algebraic valuations and direct limits

In this last section, we give an application for Corollary 4.11. We start defining the concepts of valuation-transcendental and valuation-algebraic valuations. Definition 6.1. A valuation ν on K[x] extending ν 0 is called value-transcendental if either it is not Krull or the quotient group ν(K[x])/ν 0 K is not a torsion group. We say that ν is residue-transcendental if it is Krull and the field extension K(x)ν | Kν 0 is transcendental, where we denote also by ν the unique extension of ν from K[x] to K(x). Definition 6.2. A valuation ν on K[x] extending ν 0 is called valuation-transcendental if it is value-transcendental or residue-transcendental. We say that ν is valuation-algebraic if it is not valuation-transcendental.


Siriporn Kosawatthanakun

Edwin B Clatworthy

Sajjad Ghojavand

Narongrit Sosa

Jatuporn Wittayakun

Svetlana Mintova

Application of BPH zeolite for the transesterification of glycerol to glycerol carbonate: effect of morphology, cation type and reaction condition

Keywords: nanozeolite, BPH zeolite, transesterification, glycerol, glycerol carbonate

ABSTRACT:

The valorization of glycerol (Gly) to high-value chemicals is the key to unlocking economic and sustainable biodiesel production. Transesterification of Gly with dimethyl carbonate (DMC) to glycerol carbonate (GC) requires a catalyst with high basicity and uniform pore size for high product selectivity. BPH-type zeolite (Linde Q) nanosheets are promising catalysts for overcoming diffusion transport limitations inside the pore network and inhibiting the formation of glycidol as a side product. As-prepared BPH nanosheets (nano-sized CsBPH_AP, 192 nm size) and micron-sized BPH (micron-sized KBPH_AP, 600 nm size) were prepared by hydrothermal treatment with different Si/Al ratios. With a morphological-dependent reaction, nano-sized CsBPH_AP demonstrates superior performance in the transesterification of Gly to GC than micron-sized KBPH_AP. Ion-exchange of the nano-sized CsBPH_AP and micron-sized BPH_AP, with potassium and cesium precursors respectively, to afford nano-sized K-CsBPH_IE and micron-sized Cs-KBPH_IE resulted in similar thermal properties but different basicity and catalytic activity compared to the parent samples; higher basicity resulted in better catalytic performance. The nano-sized CsBPH_AP with a catalyst loading of 6 wt% shows good catalytic performance over four runs under optimized reaction conditions at 120 ºC for 3 h and Gly:DMC molar ratio of 1:5. This study reveals the crucial effect of catalyst morphology and size on the conversion of Gly to GC.

INTRODUCTION

Biodiesel has emerged as an attractive alternative to petroleum-derived fuels because of its production from renewable feedstocks such as animal and plant fats, and its potential to offset anthropogenic CO 2 emissions from non-renewable fossil fuels. [START_REF] Mathew | Recent advances in biodiesel production: Challenges and solutions[END_REF] However, while the transesterification of glyceride and methanol produces three moles of biodiesel (fatty acid methyl esters), it also produces one mole of glycerol (Gly). [START_REF] Dodekatos | Recent Advances in Thermo-, Photo-, and Electrocatalytic Glycerol Oxidation[END_REF] As a result of increased biodiesel production there have significant efforts to valorize Gly into high value-added chemicals such as acrolein by dehydration, [START_REF] Qureshi | Highly active and selective nano H-ZSM-5 catalyst with short channels along b-axis for glycerol dehydration to acrolein[END_REF] glycerol carbonate (GC) by transesterification, [START_REF] She | Layered double hydroxide uniformly coated on mesoporous silica with tunable morphorlogies for catalytic transesterification of glycerol with dimethyl carbonate[END_REF] and 1,3-propanediol by dehydroxylation. [START_REF] Tan | Glycerol production and its applications as a raw material: A review[END_REF] In particular, GC has received increasing attention due to its versatility as a platform molecule; it can be produced by the transesterification of Gly with dimethyl carbonate (DMC).

Since DMC can serve as both a methylating agent and a solvent the reaction can be performed with or without a solvent. The reaction has been investigated in heterogeneous catalysis settings employing zeolites, [START_REF] Khanday | Biodiesel byproduct glycerol upgrading to glycerol carbonate over lithium-oil palm ash zeolite[END_REF][START_REF] Algoufi | Synthesis of glycerol carbonate by transesterification of glycerol with dimethyl carbonate over K-zeolite derived from coal fly ash[END_REF][START_REF] Pan | Transesterification of glycerol with dimethyl carbonate to glycerol carbonate over Na-based zeolites[END_REF] micro-mesoporous silicas, [START_REF] Qureshi | Highly active and selective nano H-ZSM-5 catalyst with short channels along b-axis for glycerol dehydration to acrolein[END_REF][START_REF] She | Layered double hydroxide uniformly coated on mesoporous silica with tunable morphorlogies for catalytic transesterification of glycerol with dimethyl carbonate[END_REF] silicates, [START_REF] Wang | Synthesis of glycerol carbonate from glycerol and dimethyl carbonate catalyzed by calcined silicates[END_REF] titanosilicate, [START_REF] Xiang | Transition metal-promoted hierarchical ETS-10 solid base for glycerol transesterification[END_REF] and hydrotalcites. [START_REF] Liu | Promotional effect of transition metal doping on the basicity and activity of calcined hydrotalcite catalysts for glycerol carbonate synthesis[END_REF] Although these catalysts demonstrate high performance they suffer from limitations including the molecular diffusion of the reactants and products, and reduced accessibility of basic sites. [START_REF] She | Layered double hydroxide uniformly coated on mesoporous silica with tunable morphorlogies for catalytic transesterification of glycerol with dimethyl carbonate[END_REF] To address these issues, modification of the catalyst properties such as the zeolite pore diameter, particle size, and shape offers potential solutions.

Both zeolite-based and mesoporous catalysts have been investigated for GC production.

Micron-sized chabazite (Si/Al = 2.4) in potassium form (K-CHA) consisting of aggregated quadratic prism-like crystals hydrothermally synthesized from coal fly ash gave a high yield of GC (96%) in solvent-free conditions without the leaching of metallic species (K, Ca, Ti, and Fe). [START_REF] Algoufi | Synthesis of glycerol carbonate by transesterification of glycerol with dimethyl carbonate over K-zeolite derived from coal fly ash[END_REF] However, glycidol side product was formed due to the strong basicity of the catalyst. [START_REF] Algoufi | Synthesis of glycerol carbonate by transesterification of glycerol with dimethyl carbonate over K-zeolite derived from coal fly ash[END_REF] Although the pore size of chabazite (8-ring, 0.38 × 0.38 nm) is smaller than the sizes of the reactants and product (the geometric dimensions of Gly: 0.47 × 0.52 nm, DMC: 0.37 × 0.45 nm, GC: 0.33 × 0.65 nm), [START_REF] Pan | Transesterification of glycerol with dimethyl carbonate to glycerol carbonate over Na-based zeolites[END_REF] the presence of mesopores could facilitate access to the active sites. The performance of Na-containing zeolites with different pore networks (Scheme 1) including 3A (8ring, 0.31 × 0.35 nm), 4A (10-ring, 0.41 × 0.41 nm), NaZSM-5 (10-ring, 0.51 × 0.55 and 0.53 × 0.56 nm), NaBEA (12-ring, 0.66 × 0.67 and 0.56 × 0.56 nm), and NaY (12-ring, 0.74 × 0.74 nm) have also been investigated for catalyzing the production of GC from Gly and DMC. Those zeolites were micron-sized and the reaction was performed in methanol under reflux. [START_REF] Pan | Transesterification of glycerol with dimethyl carbonate to glycerol carbonate over Na-based zeolites[END_REF] Both NaBEA and NaY were the only Na-containing zeolites reported to demonstrate the conversion of Gly to GC with 100% selectivity, of which NaY exhibited the best performance. This suggests that zeolites with large pores are preferable for the selective transesterification of Gly due to the geometric dimensions of Gly, DMC and GC. [START_REF] Pan | Transesterification of glycerol with dimethyl carbonate to glycerol carbonate over Na-based zeolites[END_REF] In addition to the size of the zeolite pores, the shape and size of the zeolite particles have been reported as significant factors in the conversion of Gly. By tuning the length of the crystal b-axis (60-70 nm), microporous H-ZSM-5 (Si/Al = 25-125) was utilized for Gly dehydration to acrolein. [START_REF] Qureshi | Highly active and selective nano H-ZSM-5 catalyst with short channels along b-axis for glycerol dehydration to acrolein[END_REF] Because of the shortened b-axis by an order of magnitude and thus the more rapid diffusion of molecules within the pore network, the catalyst exhibited a higher Gly conversion compared to a commercial ZSM-5 with a b-axis length of 1.5-1.7 µm. In addition to zeolites, mesoporous silicas containing layered double hydroxides (LDH) with three different morphologies (rice-shaped, short rods, and long rods), were proposed as nanosized composites for the transesterification of Gly with DMC in dimethyl formamide (DMF). [START_REF] She | Layered double hydroxide uniformly coated on mesoporous silica with tunable morphorlogies for catalytic transesterification of glycerol with dimethyl carbonate[END_REF] The rice-shaped catalyst with relatively shorter mesoporous channels improved the internal diffusion path lengths of both the reactants and products. Moreover, the relatively higher specific surface area of the rice-shaped nanosized composite allowed for a high density of surface basic sites, resulting in superior catalytic activity compared to the rod-shaped catalysts. This demonstrates how facilitating rapid and facile access to the basic sites by controlling the catalyst morphology is critical to achieving both high Gly conversion and GC selectivity. Furthermore, catalyst nanostructuring to achieve improved access to active sites and enhanced molecular diffusion can provide additional strategies for tailoring the catalyst behavior for Gly conversion reactions. Sulfonated porous carbons prepared from silica/resin nanocomposites exhibited greater catalytic performance compared to conventional solid acid catalysts for Gly acetylization. Despite having a lower acid site concentration the hollow sphere morphology of the sulfonated porous carbons afforded superior mass transfer properties compared to the conventional catalysts. [START_REF] Wang | Sulfonated hollow sphere carbon as an efficient catalyst for acetalisation of glycerol[END_REF] Concerning mesoporosity, the introduction of hierarchical structure to ZSM-5 zeolite catalysts and the effects on the catalytic performance on the dehydration of Gly to acrolein have shown that the type of mesoporosity formed significantly influences the catalytic performance. [START_REF] Zhang | Dehydration of Glycerol to Acrolein over Hierarchical ZSM-5 Zeolites: Effects of Mesoporosity and Acidity[END_REF] The sample with open and interconnected mesopores demonstrated better catalytic performance compared to the sample with closed and small mesopores, attributed to differences in their mass transfer properties and in-pore condensation of the reactant.

To achieve the targeted conversion of Gly to GC using environment-friendly catalysis, intrinsically basic nanosized zeolites with specific morphology present as attractive candidates as catalysts. At present, there is no report on the utilization of BPH zeolite nanosheets (Linde Q) for Gly conversion. The low-silica nanosheets possesses an appropriately-sized 12-membered ring (12MR) pore aperture along the c-axis of 0.63 × 0.63 nm, which is greater than the geometric diameters of Gly, DMC and GC (Scheme 1). [START_REF] Moliner | Multipore zeolites: synthesis and catalytic applications[END_REF] The 12MR channel is perpendicularly intersected by three eight-membered rings channels. [START_REF] Andries | The crystal structure of zeolite Linde Q: A proposal based on powder X-ray diffraction and 27Al and 29Si MAS n.m[END_REF] Initially, conventional micron-sized BPH zeolite, prepared from a K + -containing alkaline aluminosilicate precursor mixture, was not considered for catalytic applications due to its low thermal stability as a result of its high framework aluminum content (Si/Al = 1). [START_REF] Andries | The crystal structure of zeolite Linde Q: A proposal based on powder X-ray diffraction and 27Al and 29Si MAS n.m[END_REF]16 However, by employing organic templates during the synthesis procedure followed by post-synthetic treatments, higher silica-containing (Si/Al ratio up to 10) and acidic variants of BPH zeolite (UZM-4) were produced. [START_REF] Blackwell | Open-framework materials synthesized in the TMA+/TEA+ mixed-template system: the new low Si/Al ratio zeolites UZM-4 and UZM-5[END_REF][18][19] The post-synthetically treated zeolites were shown to be active for heptane cracking between 450 and 550 °C. In this work, the BPH nanosheets are prepared without organic structure-directing agents under mild (60 ºC) conditions with a relatively short synthesis time, and do not require post-synthetic template removal by calcination. [START_REF] Clatworthy | Room-Temperature Synthesis of BPH Zeolite Nanosheets Free of Organic Template with Enhanced Stability for Gas Separations[END_REF] Importantly, as a consequence of their morphology the BPH nanosheets can offer a shorter molecular diffusion path length than the other zeolites described above. Herein, BPH zeolites with different morphologies (nanosheets and micron-sized plates) were studied in terms of the correlation of their morphologies and sizes with their catalytic performance. Their intrinsic basic properties were adjustable by ion-exchanging with alkali metal cations. Their catalytic activities were investigated in the transesterification of Gly with DMC without the use of an additional solvent. The reaction factors affecting the catalytic performance including the catalyst morphology, reactant mole ratios, catalyst loading, reaction temperature and time, and reusability were explored.

EXPERIMENTAL SECTION

Chemicals and Materials

All reagents and solvents were used as received without any further purification:

aluminum hydroxide (Al(OH) 3 ,

Synthesis of nano-sized BPH zeolite

Nano-sized BPH zeolite was synthesized according to Clatworthy et al. [START_REF] Clatworthy | Room-Temperature Synthesis of BPH Zeolite Nanosheets Free of Organic Template with Enhanced Stability for Gas Separations[END_REF] with the molar oxide composition of 0.3 Cs 2 O: 1.25 K 2 O: 8.07 Na 2 O: 10 SiO 2 : 0.8 Al 2 O 3 : 120 H 2 O. The alkali aluminate mixture was prepared by dissolving 0.518 g of sodium aluminate and 3.78 g of double distilled water, followed by mixing 1.78 g of sodium hydroxide, 0.495 g of potassium hydroxide, and 0.506 g of cesium hydroxide solution (50 wt% Cs) in a sealed 60 mL polypropylene bottle.

The aluminate solution was rapidly stirred for 2 h at room temperature until a clear solution was obtained. After that, 4.77 g of colloidal SiO 2 was added dropwise with vigorous stirring to the clear aluminate solution, resulting in a turbid mixture. The mixture was vigorously stirred for 24 h followed by hydrothermal treatment at 60 ºC for 16 h. The zeolite product was centrifuged at 20,000 rpm for 20 min, washed with double distilled water until the pH of the supernatant was ~7-8, and dried in an oven at 60 ºC overnight. The as-prepared BPH zeolite nanosheets are denoted as nano-sized CsBPH_AP.

Synthesis of micron-sized BPH (Linde Q)

The molar oxide composition for the synthesis of the micron-sized BPH was: 16 K 2 O: 4 [START_REF] Clatworthy | Room-Temperature Synthesis of BPH Zeolite Nanosheets Free of Organic Template with Enhanced Stability for Gas Separations[END_REF] Potassium silicate solution was prepared by adding 6.47 g of potassium silicate to 23.92 g of double-distilled water. Then, the mixture was stirred overnight, followed by heating at 60 ºC for 30 min. The potassium aluminate solution was prepared by dissolving 3.00 g of aluminum hydroxide and 35.92 g of potassium hydroxide in 42.60 g of double distilled water, followed by heating at 90 ºC for 30 min. After allowing each solution to cool to room temperature, the potassium aluminate solution was slowly added to the potassium silicate solution with rapid stirring to form a cloudy gel. The gel was treated hydrothermally at 50 ºC for 162 h. The product was collected and washed by centrifugation at 20,000 rpm for 20 min with double distilled water until the pH of the supernatant was ~7-8. The product was dried in an oven at 60 ºC overnight. The as-prepared micron-sized BPH zeolite is denoted micronsized KBPH_AP.

SiO 2 : 1 Al 2 O 3 : 192 H 2 O.

Ion-exchange of BPH zeolite

Nano-sized CsBPH_AP (5 wt% solid) was ion-exchanged with 25 g of potassium solution (pH 9.8) containing 5 mM mixed solution of KNO 3 and KOH (10:1 w/w). For the micron-sized BPH_AP (5 wt% solid), cesium solution (CsNO 3 :CsOH = 10:1 w/w) was used.

Each mixture was stirred at room temperature for 1 h, washed with double distilled water and centrifuged at 20,000 rpm for 20 min until the pH of the supernatant was ~7-8. This procedure was repeated twice and the samples dried at 60 °C overnight. The ion-exchanged nano-sized CsBPH_AP and micron-sized BPH_AP zeolite are named nano-sized K-CsBPH_IE and micronsized Cs-KBPH_IE, respectively.

Catalyst characterization

The crystalline structure of the samples was determined by powder X-ray diffraction (XRD) with a PANalytical X'Pert PRO-MPD diffractometer with Cu Kα radiation (λ = 1.5418 Å). All patterns were collected across a 2θ range of 4-50º with a step size of 0.0167º and time per step of 950 s. The particle morphology of the zeolite samples was analyzed by field emission scanning electron microscopy (FE-SEM) using a Zeiss AURIGA FE-SEM/FIB/EDX with an upper electron detector operating at an accelerated voltage of 3 kV with a working distance of 7.1 mm. Each sample was placed on carbon tape adhered to a copper stub. The sample was coated with gold using an operating current at 10 mA under an argon atmosphere for 5 min. The particle size distribution was processed by ImageJ software version 1.49. Zeolite samples were analyzed by Fourier-transform infrared spectroscopy (FTIR) with a microscope on a Bruker Tensor27-Hyperion using ATR mode with a resolution of 4 cm -1 and 64 scans, samples were dried at 90 °C overnight before analysis. Thermogravimetric analysis was performed by a Setsys SETARAM analyzer. The samples (5-10 mg) were placed in an alumina crucible and heated from 30-800 ºC with a heating rate of 5 ºC•min -1 under a dry air flow of 40 mL•min -1 . The chemical composition of the samples was determined by inductively coupled plasma mass spectrometry (ICP-MS) using a 7900 ICP-MS from Agilent Technologies.

The basicity of the catalysts was investigated by temperature-programmed desorption of carbon dioxide (CO 2 -TPD) using a BELCAT-B chemisorption analyzer. A 50 mg sample was packed into a tubular glass reactor and pretreated at 150 ºC under helium gas flow at 30 mL⋅min -1 for 180 min to eliminate physisorbed species. Then, the sample was cooled down to 50 ºC and a gas mixture of 10% CO 2 /He was adsorbed onto the catalyst surface for 90 min. The sample was purged with He and held for 30 min to remove non-adsorbed CO 2 . The desorbed CO 2 was quantified by the TPD process in the temperature range from 50 to 150 ºC with a ramp of 5 ºC⋅min -1 and held at 150 ºC for 120 min with a flow rate of 30 mL⋅min -1 . The basicity was obtained by integration of the peak area. In addition, the basic strength was analyzed by Hammett indicator. [START_REF] Xie | Alumina-supported potassium iodide as a heterogeneous catalyst for biodiesel production from soybean oil[END_REF][START_REF] Arora | One-pot synthesis of glycerol carbonate from glycerol using three-dimensional mesoporous silicates of K/TUD-1 under environmentally benign conditions[END_REF] The indicators were bromothymol Blue (H_= 7.2) and phenolphthalein (H_= 9.8). Approximately 300 mg of sample was mixed with 10 mL of methanol containing 1 mL of the Hammett indicator and stirred for 2 hours to reach equilibrium. The color change of the solution was observed by the naked eye. The basic strengths are described as stronger than the weakest indicator that provides a color change, but weaker than the strongest indicator that makes no color change.

N 2 sorption analysis of samples was determined using Micromeritics 3Flex physisorption instrument at -196 °C. The fresh and reused samples were degassed at 300 ºC under vacuum for 8 h, nano-sized K-CsBPH_IE was degassed at 350 ºC under vacuum for 8 h, and micron-sized BPH zeolite were degassed at 120 ºC under vacuum for 24 h. The data were processed with 3Flex version 5.02 software. [START_REF] Mozgawa | FT-IR studies of zeolites from different structural groups[END_REF] Al MAS NMR spectra were recorded on a Bruker Advance III-HD 500 (11.7 T) spectrometer using 4 mm-OD zirconia rotors. [START_REF] Mozgawa | FT-IR studies of zeolites from different structural groups[END_REF] Al MAS NMR was done with a π/12 pulse (selective pulse), a spinning speed of 14 kHz and a recycle delay of 1 s. Prior to measurement, the samples were fully hydrated in a water vapor-saturated chamber at ambient temperature.

Al(NO 3 ) 3 (1 M) was used as a chemical shift reference for [START_REF] Mozgawa | FT-IR studies of zeolites from different structural groups[END_REF] Al nuclei. All spectra were normalized.

Transesterification of glycerol with dimethyl carbonate

Transesterification of glycerol (Gly) and dimethyl carbonate (DMC) to yield glycerol carbonate (GC) was performed by adding 1.50 g (16.3 mmol) of glycerol, 4.40 g (48.8 mmol) of DMC, and 0.12 g of catalyst (8 wt% compared to the amount of glycerol used) in a heavy wall pressure vessel at atmospheric pressure. The catalyst screening was carried out at 120 ºC in a silicone oil bath for 3 h, stirring at 800 rpm. The best catalyst, under the aforementioned conditions, was studied to determine the optimum reaction conditions including reaction temperature, time, catalyst loading, Gly:DMC ratio, and catalyst recyclability. After the reaction the product mixture was separated from the catalysts by centrifugation. The separated catalyst was washed with ethanol and double distilled water, dried in an oven at 80 ºC overnight and used for the next run.

The products were determined by high-performance liquid chromatography (1260 Infinity II LC System, Agilent) equipped with an Aminex HPX-87H column and a refractive index detector. An aqueous H 2 SO 4 solution (5 mM) was the mobile phase with a flow rate of 0.60 mL•min -1 and a column temperature of 55 ºC. When Gly is the limiting reactant, Gly conversion, GC yield, and selectivity of GC are calculated using equations 1, 2 and 3, respectively.

(

(2)

(3)

RESULTS AND DISCUSSION

Characterization of the catalysts

XRD patterns of the nano-and micron-sized BPH zeolite samples are shown in Fig. 1.

All samples possess the BPH framework topology. The diffraction patterns of the nano-sized CsBPH_AP have relatively less intensity and broader peaks than the micron-sized BPH due to the smaller crystalline domains [START_REF] Clatworthy | Room-Temperature Synthesis of BPH Zeolite Nanosheets Free of Organic Template with Enhanced Stability for Gas Separations[END_REF] ; this is further confirmed by SEM analysis (vide infra). In particular, the (001) reflection of the nanosheets at 2θ of ≈ 6.4º has a very low intensity due to the significantly reduced sheet thickness compared to the micron-sized BPH sample. The XRD patterns of the ion-exchanged samples are not significantly different from those of the parent samples. This indicates that the crystal structure and the morphology of the samples are preserved during the ion-exchange process. SEM images of the nano-and micron-sized BPH zeolite samples are presented in Fig. 2.

The nano-sized CsBPH_AP sample is described by agglomerated nano-sized sheets with dimensions of 192±15 nm across and 20±3 nm thick. The morphology is similar to the nanosized BPH zeolite reported by Clatworthy et al. [START_REF] Clatworthy | Room-Temperature Synthesis of BPH Zeolite Nanosheets Free of Organic Template with Enhanced Stability for Gas Separations[END_REF] The micron-sized KBPH_AP is described by plate-like particles 541±40 nm across and 57±9 nm thick. The SEM analysis of the BPH zeolite samples demonstrates the different sizes of the zeolite particles which could play a role in determining their catalytic performance. A smaller particle size generally facilitates more rapid diffusion of the reactants and products through the zeolite pore network. From the ICP analysis presented in Table 1, the sodium content from the nano-sized K-CsBPH_IE slightly decreased after ion exchange with potassium solution. Meanwhile, the cesium amount from micron-sized Cs-KBPH_IE increased after ion exchange with cesium solution. Although the BPH-type zeolite nanosheets were ion exchanged with a different alkali metal solutions, the original type of alkali metal cations (Na + , K + , Cs + ) were preserved, in part due to their strong ionic interaction with the zeolite structure. [START_REF] Yang | CO 2 adsorption over ion-exchanged zeolite beta with alkali and alkaline earth metal ions[END_REF][START_REF] Walton | CO 2 adsorption in Y and X zeolites modified by alkali metal cation exchange[END_REF] The nano-sized CsBPH_AP sample has a higher Si/Al molar ratio than the micron-sized KBPH_AP which is likely a reflection of the different Si and Al contents of the synthesis recipies. After the ion-exchange process the Si/Al ratio of the ion-exchanged zeolites was found to be unchanged compared to the parent samples. The FTIR spectra of the as-prepared and ion-exchanged BPH zeolites are presented in Fig. 3. The nano-sized CsBPH_AP and micron-sized KBPH_AP have similar FTIR spectra but different peak positions due to their different Si/Al ratios. [START_REF] Ma | Facile and fast determination of Si/Al ratio of zeolites using FTIR spectroscopy technique[END_REF] The characteristic bands of BPH zeolite at 1150-950 cm -1 and 740-660 cm -1 are attributed to the asymmetric and symmetric stretching of the zeolite framework, respectively. The bands at 496, and 415-410 cm -1 are assigned to the bending mode of Si-O-T, where T is a framework Si or Al atom. [START_REF] Andries | The synthesis and properties of synthetic zeolite Linde Q[END_REF] An intense band at 598-588 cm -1 is typically associated with double rings. Although BPH does not possess double rings, this band could be ascribed to the vibration of six rings in the afs and bph composite building units. [START_REF] Mozgawa | FT-IR studies of zeolites from different structural groups[END_REF] Moreover, the bands from OH groups or water are observed at 1649 and 3400 cm -1 . The FTIR spectra of the nano-sized K-CsBPH_IE and micron-sized Cs-KBPH_IE are similar to their parent zeolites; no significant differences are observed. The thermal properties of the BPH zeolite samples determined by thermogravimetric analysis in air are presented in Fig. 4 and are similar to those previously reported under vacuum. [START_REF] Clatworthy | Room-Temperature Synthesis of BPH Zeolite Nanosheets Free of Organic Template with Enhanced Stability for Gas Separations[END_REF] The nano-sized CsBPH_AP sample displays significant mass loss below 130 °C due to the removal of surface and physisorbed water; further mass loss to ~300 °C is due to the removal of cation-coordinated water. In comparison, the micron-sized KBPH_AP sample exhibits a lower degree of mass loss below 110 ºC followed by a second event at 220 ºC. The The CO 2 -TPD profiles of the as-prepared and ion-exchanged BPH zeolites are shown in Fig. 5 and the basicities are listed in Table 1. Micron-sized KBPH_AP and nano-sized CsBPH_AP show a desorption peak at 93 ºC and 94 ºC, respectively, indicating no difference in interaction with the probe molecule CO 2 . However, micron-sized KBPH_AP shows stronger intrinsic basicity because of a lower Si/Al ratio. [START_REF] Al-Ani | Nanostructured zeolites: the introduction of intracrystalline mesoporosity in basic faujasite-type catalysts[END_REF][START_REF] Busca | Acidity and basicity of zeolites: A fundamental approach[END_REF][START_REF] Beltrao-Nunes | CO 2 capture by coal ash-derived zeolites-roles of the intrinsic basicity and hydrophilic character[END_REF] After ion exchange, the CO 2 -TPD profiles of the micron-sized Cs-KBPH_IE and nano-sized K-CsBPH_IE exhibit slight changes compared to their parent samples. With a higher Cs content, micron-sized Cs-KBPH_IE provides stronger basicity than micron-sized KBPH_AP. The results are consistent with the case of CO 2 adsorption on zeolite Y at low pressure, whereby the basicity of the zeolite framework oxygens decreases with decreasing size of the extra-framework cation, i.e. from Cs + to Li + . The simultaneous interaction of CO 2 with Cs + and neighboring framework oxygen atoms on CsY stabilizes CO 2 adsorption more than that on KY. [START_REF] Pirngruber | The role of the extra-framework cations in the adsorption of CO 2 on faujasite Y[END_REF][START_REF] Mortier | Zeolite electronegativity related to physicochemical properties[END_REF] However, the nano-sized K-CsBPH_IE exhibits a slightly lower basicity compared to the parent sample despite the exchange of approximately one Na + for

Catalytic activity for the transesterification of glycerol with DMC

Catalyst screening

A comparison of the BPH zeolite samples with different particle sizes (micron vs nano) and cation contents on the performance in the transesterification of Gly with DMC is presented in Fig. 6. A blank reaction without catalyst affords a poor conversion of Gly and a low yield of the GC product. This result is consistent with the literature that the reaction performed without catalyst at 70 ºC in methanol under reflux did not show Gly conversion and selectivity towards GC; in another work a small GC yield (≈1%) was detected at 100 ºC. [START_REF] Pan | Transesterification of glycerol with dimethyl carbonate to glycerol carbonate over Na-based zeolites[END_REF][START_REF] Liu | Promotional effect of transition metal doping on the basicity and activity of calcined hydrotalcite catalysts for glycerol carbonate synthesis[END_REF] The higher reaction temperature used in this work is responsible for the observed low Gly conversion and GC yield for the catalyst-free run. [START_REF] Teng | Process optimization and kinetics of microwave assisted transesterification of crude glycerol for the production of glycerol carbonate[END_REF] Considering the difference in the particle size of the BPH zeolite samples only, nanosized CsBPH_AP and nano-sized K-CsBPH_IE demonstrate better transesterification performance for the conversion of Gly and yield of GC than micron-sized KBPH_AP and micron-sized Cs-KBPH_IE. The better performance could be attributed to the shorter diffusion paths of the reagents and larger surface areas, which is the advantage of nano-over micron-sized zeolites. [START_REF] Mintova | Nanosized zeolites: Quo Vadis?[END_REF] To the best of our knowledge, this is the first report comparing nano-and micronsized zeolites for the transesterification of Gly to GC. The effect of the catalyst particle shape and size was reported by She et al. [START_REF] She | Layered double hydroxide uniformly coated on mesoporous silica with tunable morphorlogies for catalytic transesterification of glycerol with dimethyl carbonate[END_REF] The rice-shaped LDH/SBA-15 nanocomposite showed higher activity due to the greater specific surface area and reduced diffusion path length of the reactants and products compared to the other morphologies investigated.

The effect of basicity on the performance of the BPH zeolite catalysts by varying the extra-framework cation content is compared in Fig. 6. After ion exchange of the nano-sized CsBPH_AP with K + , the obtained nano-sized K-CsBPH_IE demonstrated a lower catalytic activity than the nano-sized CsBPH_AP. In contrast by introducing Cs + to the micron-sized KBPH_AP, micron-sized Cs-KBPH_IE showed a higher activity than the parent KBPH. The catalytic performances are consistent with the basicity from CO 2 -TPD (Table 1 and Figure 5).

This finding is in good agreement with the report by Al-Ani et al. [START_REF] Al-Ani | Nanostructured zeolites: the introduction of intracrystalline mesoporosity in basic faujasite-type catalysts[END_REF] on the basicity of ionexchanged FAU and performance in transesterification of rapeseed oil with methanol.

From these results, not only is the morphology of the BPH nanosheets is crucial, but the basicity is also key to optimizing the transesterification of Gly with DMC. From the results, the nano-sized CsBPH_AP catalyst was subject to further study.

Effect of different reaction parameters on the transesterification

The effect of the reaction temperature across the range of 90-135 ºC on the transesterification of Gly with DMC using the nano-sized CsBPH_AP catalyst is shown in Fig. 7a. The Gly conversion and GC yield increase with the temperature, indicating that the reaction is endothermic. A similar effect was observed from the reaction catalyzed by transition-metal doped hydrotalcite [START_REF] Liu | Promotional effect of transition metal doping on the basicity and activity of calcined hydrotalcite catalysts for glycerol carbonate synthesis[END_REF] and K-CHA. [START_REF] Algoufi | Synthesis of glycerol carbonate by transesterification of glycerol with dimethyl carbonate over K-zeolite derived from coal fly ash[END_REF] The viscosity of reactants is reduced at a higher temperature, resulting in better miscibility of Gly and DMC. Although the Gly conversion and GC yield further increase at the reaction temperature of 135 ºC, the selectivity towards GC decreases due to the formation of side products. The results indicate that a reaction temperature above 120 ºC is not suitable. Accordingly, the optimal temperature for the transesterification of Gly and DMC catalyzed by nano-sized CsBPH_AP is 120 ºC. The effect of the reaction time from 1 to 6 h on the transesterification of Gly with DMC over the nano-sized CsBPH_AP catalyst is presented in Fig. 7b. The Gly conversion and the GC yield increase with the reaction time. The performance at 6 h was slightly higher than that at 3 h, indicating the equilibrium was approached. In comparison with the literature, the time to reach the equilibrium varied with the reaction conditions such as temperature, type of reactor, mole ratio of reactants, and type and loading of catalyst. For example, to reach equilibrium over NaY under reflux at 70 ºC required 4 h, 8 but 1.5 h over KCHA at 75 ºC. [START_REF] Algoufi | Synthesis of glycerol carbonate by transesterification of glycerol with dimethyl carbonate over K-zeolite derived from coal fly ash[END_REF] The longer reaction time might lead to a decrease in the GC selectivity due to the formation of side products. [START_REF] Algoufi | Synthesis of glycerol carbonate by transesterification of glycerol with dimethyl carbonate over K-zeolite derived from coal fly ash[END_REF] Considering the significant increase of the reaction activity with time, the duration of 3 h is suitable for further evaluation.

The comparison of catalyst loading across the range of 4-12 wt% on the transesterification of Gly with DMC using the nano-sized CsBPH_AP catalyst is shown in Fig. 7c. Both Gly conversion and GC yield increase dramatically with the loading from 4 to 6 wt%.

This could be from the increase in the concentration of active basic sites that initiate Gly transesterification by abstracting its proton from the first hydroxyl group. [START_REF] Algoufi | Synthesis of glycerol carbonate by transesterification of glycerol with dimethyl carbonate over K-zeolite derived from coal fly ash[END_REF] The catalyst loading beyond 6 wt% results in the decline of activity in a linear manner. This is possibly due to the increased viscosity of the mixture and the adsorption of the reactants and products on the catalyst surface. In addition, the change in catalyst loading from 4 to 12 wt% does not affect the selectivity of the GC product. From these results, the optimal catalyst loading is 6 wt%.

The comparison of the molar ratios of glycerol to DMC on the transesterification is presented in Fig. 7d. At the equivalent molar ratio, both Gly conversion and GC yield are below 50 mol% due to the increased viscosity and poor mass transfer of reactants in the reaction system. At a molar ratio of glycerol to DMC above 1:3, the reaction activity increases because the higher molar ratio could shift the reaction equilibrium towards the GC product. [START_REF] Algoufi | Synthesis of glycerol carbonate by transesterification of glycerol with dimethyl carbonate over K-zeolite derived from coal fly ash[END_REF][START_REF] Arora | One-pot synthesis of glycerol carbonate from glycerol using three-dimensional mesoporous silicates of K/TUD-1 under environmentally benign conditions[END_REF][START_REF] Das | A green and facile production of catalysts from waste red mud for the one-pot synthesis of glycerol carbonate from glycerol[END_REF] Considering the reaction setup, the Gly:DMC molar ratio of 1:5 facilitates the magnetic stirring by decreasing the mixture viscosity. Finally, the molar ratio of 1:9 provides an excess amount of DMC that does not affect the reaction activity. The Gly conversion and GC products are slightly increased.

The results from the catalytic testing demonstrate that the nano-sized CsBPH_AP, identified as the best catalyst in this work from the optimal reaction conditions, has potential as a catalyst for the transesterification of Gly among the catalysts already reported in the literature (Table S2).

Reusability of the catalyst

The reusability and leaching tests of the nano-sized CsBPH_AP catalyst for the transesterification of Gly with DMC with the optimal reaction conditions are shown in Fig. 8a.

The nano-sized CsBPH-AP retains its initial catalytic performance in the first two runs followed by a decrease of the Gly conversion and GC yield by 14% and 13%, respectively. However, the GC selectivity remains constant across all four runs. The results show that the nano-sized CsBPH_AP catalyst is susceptible to activity loss during transesterification. The leaching test was performed to investigate the catalyst stability by separating the nano-sized BPH zeolite catalyst by hot filtration after 1 h of reaction while the filtrate was kept under the same reaction conditions without the catalyst for another 2 h. The analysis of the reaction mixture revealed no further glycerol conversion after the removal of the nano-sized BPH zeolite catalyst, demonstrating that leaching was not the cause of deactivation (Fig. 8b). The nano-sized CsBPH_AP catalyst was analyzed after the fourth run by XRD, FTIR, TGA, N 2 sorption, [START_REF] Mozgawa | FT-IR studies of zeolites from different structural groups[END_REF] Al MAS NMR and CO 2 -TPD. The results from each technique were compared with that of the fresh catalyst in Fig. 9. The reused and fresh nano-sized CsBPH_AP show similar results from XRD, FTIR, TGA, and N 2 sorption (Fig. 9a-d, respectively). In addition, ICP-MS analysis of the nano-sized Cs-BPH_AP catalyst after the fourth reaction cycle revealed negligible differences in the composition between the fresh ((Cs TPD profiles and slightly lower intensity in [START_REF] Mozgawa | FT-IR studies of zeolites from different structural groups[END_REF] Al NMR spectra (Fig. 9e-f, respectively). The results suggest that the cause of deactivation could be dealumination. As a consequence, further investigation to understand the changes during the reaction is necessary to find ways to retain the catalyst performance over a greater number of runs. Smaller particles, specifically a reduction in the length along the c-axis, allows for better diffusion of the reactants and products through the zeolite pore network. Where catalytsts with similar morphologies and particle sizes are compared, the zeolite with the higher basicity displays better performance.

The optimum reaction condition of the best catalyst, nano-sized CsBPH_AP is 120 ºC for 3 h with a catalyst loading of 6 wt % and Gly:DMC molar ratio of 1:5 svetlana.mintova@ensicaen.fr

Notes

The authors declare no competing financial interest.

Scheme 1 .

 1 Scheme 1. Comparison of the sizes of DMC, Gly and GC, and the comparison of the zeolite

Fig. 1

 1 Fig. 1 XRD patterns of (a) nano-sized CsBPH_AP (black) and nano-sized K-CsBPH_IE (blue)

Fig. 2

 2 Fig. 2 SEM images of (a) nano-sized CsBPH_AP, (b) nano-sized K-CsBPH_IE, (c) micron-sized

Fig. 3

 3 Fig. 3 FTIR spectra of (a) nano-sized CsBPH_AP, (b) nano-sized K-CsBPH_IE, (c) micron-sized

  difference in the mass loss behavior is related to the difference in the zeolite external surface areas, diffusion path length of water through the zeolites, and extra-framework cation compositions. At 350 ºC both samples show complete removal of the physisorbed and chemisorbed water. Both the potassium-exchanged nano-sized K-CsBPH_IE and cesiumexchanged micron-sized Cs-KBPH_IE exhibit a similar mass loss at 350 °C compared to their parent samples (16.3 vs 15.6% and 15.1 vs 15.7% respectively) due to their similar chemical compositions.

Fig. 4

 4 Fig. 4 TG and dTG profiles of the BPH zeolite samples (a) nano-sized CsBPH_AP (black) and

Fig. 5

 5 Fig. 5 CO 2 -TPD profiles of (a) nano-sized CsBPH_AP (black) and nano-sized K-CsBPH_IE

Fig. 6

 6 Fig. 6 Comparison of the as-prepared and ion-exchanged micron-sized and nano-sized BPH

Fig. 7

 7 Fig. 7 Effects of different reaction parameters on the transesterification of Gly with DMC using

Fig. 8

 8 Fig. 8 (a) Reusability and (b) leaching tests of the nano-sized CsBPH_AP catalyst under the

Fig. 9

 9 Fig. 9 Comparison between the fresh nano-sized CsBPH_AP and the reused sample after the

  extra pure, Acros Organics), LUDOX ® AS-40 colloidal silica (SiO 2 , Aldrich, 40 wt% suspension), cesium chloride (CsCl, 99.9%, Alfa Aesar), dimethyl carbonate (99%, Alfa Aesar), glycerol (C 3 H 8 O 3 , 99.5%, Alfa Aesar) potassium silicate (K 2 SiO 3 , anhydrous ~48 Mesh, SiO 2 :K 2 O 2.5:1 wt%, Alfa Aesar), ethanol (C 2 H 5 OH, 99.9%, Carlo Erba), methanol (CH 3 OH, 99%, Carlo Erba), cesium nitrate (CsNO 3 , 99%, Sigma Aldrich), glycerol carbonate (C 4 H 6 O 4 , 90%, Sigma Aldrich), glycidol (C 3 H 6 O 2 , ≥95.0%, Sigma Aldrich), sulfuric acid (H 2 SO 4 , 98%), potassium hydroxide (KOH, 90%, Sigma Aldrich), potassium nitrate (KNO 3 , 99%, Sigma Aldrich), sodium aluminate (NaAlO 2 , 50-56% Al 2 O 3 , 40-45% Na 2 O, Sigma-Aldrich), and sodium hydroxide (NaOH, 99%, VWR Chemicals). An aqueous solution of cesium hydroxide (CsOH, 50 wt% Cs) was prepared by dissolving cesium hydroxide hydrate (CsOH•xH 2 O (15-20% H 2 O), 99.9%, Alfa Aesar) in double-distilled water. All water used was double distilled using Aquatron water still A4000D.

Table 1 .

 1 Elemental composition of BPH-type zeolite samples determined by ICP-MS, and basicity measured by CO 2 -TPD.

	Catalyst	Composition	Si/Al ratio (mol)	Basicity (mmol CO 2 •g -1 )
	micron-sized KBPH_AP	(K 14 )Si 14 Al 14 O 56	1.00	0.327
	micron-sized Cs-KBPH_IE	(CsK 13 )Si 14 Al 14 O 56	1.00	0.350
	nano-sized CsBPH_AP	(Cs 4.2 K 2.3 Na 5.5 )Si 16 Al 12 O 56	1.33	0.130
	nano-sized K-CsBPH_IE	(Cs 4.2 K 3.3 Na 4.5 )Si 16 Al 12 O 56	1.33	0.105

  4.2 K 2.3 Na 5.5 )Si 16 Al 12 O 56 ) and used ((Cs 4.2 K 2.3 Na 5.7 )Si 15.7 Al 12.2 O 56 ) catalysts. This suggests that the structure of the nanosized CsBPH_AP is stable under the catalytic test conditions, making it a good candidate for further development. However, the reused sample has a lower basicity according to the CO 2 -

  . After four runs the nanosized sample exhibits good stability in terms of structure and morphology, but a slight decrease in basicity. Overall, this work demonstrates that nano-sized CsBPH_AP is a promising catalyst Siriporn Kosawatthanakun, School of Chemistry, Institute of Science, Suranaree University of Technology, Nakhon Ratchasima 30000, Thailand, orcid.org/0000-0002-6637-6049; E-mail: kosa.siriporn@gmail.com Sajjad Ghojavand, Normandie Université, ENSICAEN, UNICAEN, CNRS, Laboratoire Catalyse et Spectrochimie (LCS), 14050 Caen, France, orcid.org/0000-0003-4136-2406; E-mail: sajjad.ghojavand@ensicaen.fr Narongrit Sosa, Nuclear Technology Research and Development Center, Thailand Institute of Nuclear Technology (Public Organization), Nakhon Nayok 26120, Thailand, orcid.org/0000-0002-7745-329X; E-mail: narongrit.sci@gmail.com Svetlana Mintova, Normandie Université, ENSICAEN, UNICAEN, CNRS, Laboratoire Catalyse et Spectrochimie (LCS), 14050 Caen, France, orcid.org/0000-0002-0738-5244; E-mail:

for the transesterification of Gly to produce GC and may find further application in other basecatalysed reactions. Graphical abstract ACKNOWLEDGMENTS Kosawatthanakun, S. is supported by the Royal Golden Jubilee Ph.D. program (Grant No. PHD/0142/2557) from the Thailand Research Fund (TRF) and The French Embassy in Thailand with contribution to the RGJ-Ph.D. program. The financial support from the Centre of zeolites and nanoporous materials, Region Normandie, CNRS (CLEAR) is acknowledged.

AUTHOR INFORMATION Corresponding Authors

Jatuporn Wittayakun -School of Chemistry, Institute of Science, Suranaree University of Technology, Nakhon Ratchasima 30000, Thailand, orcid.org/0000-0002-4426-2825; E-mail: jatuporn@g.sut.ac.th Edwin B. Clatworthy, Normandie Université, ENSICAEN, UNICAEN, CNRS, Laboratoire Catalyse et Spectrochimie (LCS), 14050 Caen, France, orcid.org/0000-0002-7204-2213; E-mail: edwin.clatworthy@ensicaen.fr

Authors