The valuation pairing on an upper cluster algebra

It is known that many (upper) cluster algebras are not unique factorization domains. We exhibit the local factorization properties with respect to any given seed $t$: any non-zero element in a full rank upper cluster algebra can be uniquely written as the product of a cluster monomial in $t$ and another element not divisible by the cluster variables in $t$. Our approach is based on introducing the valuation pairing on an upper cluster algebra: it counts the maximal multiplicity of a cluster variable among the factorizations of any given element. We apply the valuation pairing to obtain many results concerning factoriality, $d$-vectors, $F$-polynomials and the combinatorics of cluster Poisson variables. In particular, we obtain that full rank and primitive upper cluster algebras are factorial; an explanation of $d$-vectors using valuation pairing; a cluster monomial in non-initial cluster variables is determined by its $F$-polynomial; the $F$-polynomials of non-initial cluster variables are irreducible; and the cluster Poisson variables parametrize the exchange pairs of the corresponding upper cluster algebra.

Background.Around the year 2000, Fomin and Zelevinsky introduced cluster algebras [FZ02] with the aim of developing a combinatorial approach to the theory of canonical bases in quantum groups and the closely related theory of total positivity in algebraic groups.Since then, cluster algebras have been linked to numerous other subjects and their study has flourished, cf. for example the surveys [Lec10, Fom10, Kel12, KD20].A cluster algebra A is a subalgebra of an ambient field F generated by certain combinatorially defined generators (called cluster variables), which are grouped into overlapping sets (called clusters) of constant cardinality n.Different clusters are obtained from each other by a sequence of mutations.One remarkable feature of cluster algebras is the Laurent phenomenon [FZ02], that is, for any given cluster A t0 = (A 1;t0 , . . ., A n;t0 ), each cluster variable A k;t can be written as A k;t = P (A 1;t0 , . . ., A n;t0 ) , where P is a polynomial in variables from A t0 such that A i;t0 does not divide P for any i.The vector d t0 (A k;t ) = (d 1 , . . ., d n ) T is called the d-vector of A k;t with respect to A t0 and the polynomial P is called the numerator polynomial of A k;t with respect to A t0 .
For each cluster algebra A, Fock-Goncharov defined [FG09] a pair of varieties: the cluster K 2 -variety and the cluster Poisson variety (we follow the terminology of the appendix to [SW20]).The upper cluster algebra U is defined to be the algebra of global functions on the cluster K 2 -variety, and the cluster Poisson algebra X is defined to be the algebra of the global functions on the cluster Poisson variety.The clusters of U correspond to local toric charts and the cluster variables correspond to local coordinates (which happen to be global functions) of the cluster K 2 -variety.The cluster Poisson variety admits a canonical atlas of dual toric charts whose coordinates are the cluster Poisson variables (but they are often not global) of X .1.2.Main results.An upper cluster algebra is constructed from its seeds (see Section 2 for details).When the upper cluster algebra has geometric type, each seed t corresponds to an extended exchange matrix B t , see Definition 2.10.The upper cluster algebra is said to be full rank or primitive if B t is.Such properties are independent of the choice of the seed t.
Valuation pairings, factoriality and the Ray Fish Theorem.It is known from [GLS13] that many (upper) cluster algebras are not unique factorization domains.In order to study the local factorization properties of upper cluster algebras, we introduce the valuation pairing (see Definition 3.1) on any upper cluster algebra U. To each pair (A k;t , M ) consisting of a cluster variable A k;t and an element M in U, it associates the largest integer s (possibly infinity) such that M/A s k;t still belongs to U. We write (A k;t || M ) v = s.Using the valuation pairing we prove that any full rank upper cluster algebra has the following local unique factorization property: For each seed t of U, any non-zero element M can be uniquely factorized as M = N •L, where N is a cluster monomial in t and L is an element in U not divisible by any cluster variable in t.We give many applications to d-vectors, F -polynomials, factoriality of upper cluster algebras and combinatorics of cluster Poisson variables.
As an application to factoriality of upper cluster algebras, we prove that a full rank upper cluster algebra U with initial seed t 0 is factorial if and only if each exchange binomial of t 0 is irreducible in the corresponding polynomial ring (see Theorem 4.9).In particular, full rank, primitive upper cluster algebras are factorial (see Theorem 4.13 (i)).These include principal coefficient upper cluster algebras as a special case.For full rank, primitive upper cluster algebras, we also prove that the numerator polynomials of non-initial cluster variables are irreducible (see Theorem 4.13 (ii)).
The Starfish Theorem in [BFZ05], cf. also Theorem 2.16 of the present paper, plays a very important role in this paper.It states that any full rank upper cluster algebra U can be written as the intersection of n+1 Laurent polynomial rings.Here n is the rank of U. Thanks to the results on factoriality of upper cluster algebras, we show that any full rank, primitive upper cluster algebra can be written as the intersection of two Laurent polynomial rings (see Theorem 4.23).We call this the Ray Fish Theorem.
Application to d-vectors.In [CL20], Li and the first author of the present paper proved that there exists a well-defined function (− || −) d on the set of cluster variables, which is called the d-compatibility degree.The values of the d-compatibility degree (− || −) d are given by the components of the d-vectors.One remarkable property of the d-compatibility degree is that it uniquely determines how the set of cluster variables is grouped into clusters.As an application to d-vectors, we show how to express the d-compatibility degree and the d-vectors using the valuation pairing for full rank upper cluster algebras (see Theorem 5.1).As an application, in the full rank, primitive case, we prove that if M is a monomial in non-initial cluster variables, then M and its d-vector are uniquely determined by the numerator polynomial P M of M (see Proposition 5.6).
Application to F -polynomials.Let B be an n × n skew-symmetrizable integer matrix.The F -polynomial F B;t0 k;t associated with (B, t 0 ; k, t) may be constructed by an explicit recursion or using the cluster algebra A with principal coefficients [FZ07] associated with (B, t 0 ): Let A t0 = (A 1;t0 , . . ., A n;t0 ) be the initial cluster of A and A k;t a cluster variable of A. In this case, the cluster variable A k;t can be written as a Laurent polynomial in Z[Z 1 , . . ., Z n ][A ±1 1;t0 , . . ., A ±1 n;t0 ].Then the F -polynomial of A k;t is the specialization given by M i be a monomial in cluster variables, where each M i is a cluster The F -polynomials are the non tropical ingredients of the canonical expressions [NZ12] for both, the cluster variables and the cluster Poisson variables.They are fundamental in the additive categorification of cluster algebras (see for example [CC06,DWZ10] and the surveys [BM06, GLS08, Kel10, Kel12, Pla18]) and in their link to Donaldson-Thomas theory (see for example [KS08,Nag13,Bri17]).It is known that F -polynomials enjoy many nice properties, for example, they have positive coefficients and constant term 1.We refer the readers to [LS15, DWZ10, GHKK18] for these results and to [GY20,Gyo21,FG23,Fei23a,Fei23b,LP22] for some recent work on F -polynomials.
As an application to F -polynomials, we prove that if M is a monomial in noninitial cluster variables, then M is uniquely determined by its F -polynomial F M (see Theorem 6.1).We also prove that the F -polynomials of non-initial cluster variables are irreducible (see Theorem 6.2).
Application to cluster Poisson variables.As an application to combinatorics of cluster Poisson variables, we give several equivalent characterizations of when two cluster Poisson variables are equal (see Theorem 7.5).Recall that each mutation t ′ = µ k (t) gives an A -exchange pair (A k;t , A k;t ′ ) of the upper cluster algebra U and an X -exchange pair (X k;t , X k;t ′ ) of the cluster Poisson algebra X .
As the first application of Theorem 7.5, we prove that the cluster Poisson variables of a cluster Poisson algebra X parametrize the A -exchange pairs of the upper cluster algebra U of the same type as X (see Theorem 7.6).This extends the corresponding result by Sherman-Bennett [SB19] from the finite type case to full generality.
As the second application of Theorem 7.5, we prove that the X -seeds of X whose Poisson clusters contain particular cluster Poisson variables form a connected subgraph of the exchange graph of X (see Theorem 7.10).This is analogous to the result on exchange graphs of cluster algebras given in [CL20], cf. also Theorem 2.17 of this paper.1.3.Contents.This paper is organized as follows: In Section 2, some basic definitions, notations and known results are introduced.In Section 3, we introduce the valuation pairing (− || −) v on any upper cluster algebra and prove the local unique factorization property for full rank upper cluster algebras (see Theorem 3.7).In Sections 4, 5, 6, 7, we give the applications to d-vectors, F -polynomials, factoriality of upper cluster algebras and combinatorics of cluster Poisson variables.To be more precise: In Section 4.2, we prove that a full rank upper cluster algebra U with initial seed t 0 is factorial if and only if each exchange binomial of t 0 is irreducible in the corresponding polynomial ring (see Theorem 4.9).In particular, we show in Section 4.3 that full rank, primitive upper cluster algebras are factorial (see Theorem 4.13 (i)).Moreover, for these upper cluster algebras, we also show that the numerator polynomials of non-initial cluster variables are irreducible (see Theorem 4.13 (ii)).In Section 4.4, we give some examples of non-factorial upper cluster algebras.In Section 4.5, we prove the Ray Fish Theorem, which states that any full rank, primitive upper cluster algebra U can be written as the intersection of two Laurent polynomial rings (see Theorem 4.23).
In Section 5.1, we show how to express the d-compatibility degree and the dvectors using the valuation pairing for full rank upper cluster algebras (see Theorem 5.1).In Section 5.2, we give a local factorization for cluster monomials (see Theorem 5.4).As an application, in the full rank, primitive case, we prove that if M is a monomial in non-initial cluster variables, then M and its d-vector are uniquely determined by the numerator polynomial P M of M (see Proposition 5.6).
In Section 6.1, we prove that if M is a monomial in non-initial cluster variables, then M is uniquely determined by its F -polynomial F M (see Theorem 6.1).In Section 6.2, we prove that the F -polynomials of non-initial cluster variables are irreducible (see Theorem 6.2).
In Section 7, we give several equivalent characterizations of when two cluster Poisson variables are equal (see Theorem 7.5).As the first application, we prove that the cluster Poisson variables of a cluster Poisson algebra X parametrize the Aexchange pairs of the upper cluster algebra U of the same type as X (see Theorem 7.6).As the second application, we prove that the X -seeds of X whose Poisson clusters contain particular cluster Poisson variables form a connected subgraph of the exchange graph of X (see Theorem 7.10).
The following diagram gives the logical dependence among the proofs of the main theorems in this paper.
Theorem 4.9 4.13 o o Theorem 6.2 Theorem 6.1 / / Theorem 7.10 1.4.Convention and assumption.Throughout this article, K is assumed to be a factorial domain of characteristic 0 (e.g., K = Z, Q, R, C) and all upper cluster algebras are considered as algebras over KP, where P is some abelian multiplicative group and KP the corresponding group ring.The factoriality of K is necessary for us, because one of the aims of this paper is to consider the factoriality of upper cluster algebras.We always assume that the exchange binomials of upper cluster algebras in this article are not invertible in KP.Note that when K = Z, this condition is always satisfied.When K is a field and we have a trivial exchange relation in an upper cluster algebra U, we can always freeze the cluster variable A k and consider the upper cluster algebra U † with smaller rank over KP † = KP[A ± k ].Note that U † and U are isomorphic as K-algebras.
Acknowledgements.The authors are very grateful to Luc Pirio, whose conjectures have inspired a good part of the results of this paper.P. Cao would like to thank Xiaofa Chen, Yu Wang and Yilin Wu for helpful discussions during his stay in Paris.B. Keller is indebted to Lauren Williams and Melissa Sherman-Bennett for their interest and for stimulating conversations.The authors are grateful to Ana Garcia Elsener and Daniel Smertnig for a helpful email exchange [GES22] where, in particular, they defined the function λ (cf.also Remark 4.2 and the Appendix).The authors sincerely thank the anonymous referee for valuable comments including the suggestion of Proposition 4.6.P.
It is not hard to check that the submatrix B ′ of B ′ is still skew-symmetrizable with the same skew-symmetrizer as B and that µ k is an involution.Proposition 2.2.([BFZ05, Lemma 3.2]).Matrix mutations preserve the rank of B.
Recall that (P, ⊕, •) is a semifield if (P, •) is an abelian multiplicative group endowed with a binary operation of auxiliary addition ⊕ which is commutative, associative and satisfies that the multiplication distributes over the auxiliary addition.
The tropical semifield P = Trop(Z 1 , . . ., Z m ) is the free (multiplicative) abelian group generated by Z 1 , . . ., Z m with auxiliary addition ⊕ defined by i Let Q sf (Z 1 , . . ., Z m ) be the set of all non-zero rational functions in m independent variables Z 1 , . . ., Z m , which can be written as subtraction-free rational expressions in Z 1 , . . ., Z m .The set Q sf (Z 1 , . . ., Z m ) is a semifield with respect to the usual operations of multiplication and addition.It is called an universal semifield.Definition 2.3 (X -seed and cluster Poisson seed).(i) A (labeled) X -seed over a semifield P is a pair (B, X), where • B = (b ij ) is an n×n skew-symmetrizable integer matrix, called an exchange matrix ; • X = (X 1 , . . ., X n ) is an n-tuple of elements in P. We call X the X -cluster and X 1 , . . ., X n the X -variables of (B, X). (ii) Let (B, X) be an X -seed over a semifield P. If P = Q sf (X 1 , . . ., X n ), we call (B, X) a cluster Poisson seed, X the Poisson cluster, and X 1 , . . ., X n the cluster Poisson variables of (B, X).
Recall that K is assumed to be a factorial domain of characteristic 0. We take the ambient field F to be the field of rational functions in n independent variables with coefficients in KP.
Definition 2.4 (A -seed).A (labeled) A -seed over P is a triple (B, X, A), where • (B, X) forms an X -seed over P; • A = (A 1 , . . ., A n ) is an n-tuple such that {A 1 , . . ., A n } is a free generating set of F over KP.We call A the cluster and A 1 , . . ., A n the cluster variables of (B, X, A); Definition 2.5 (X -seed mutation and X -exchange pair).Let (B, X) be an Xseed over P. Define the X -seed mutation of (B, X) at k ∈ {1, . . ., n} as a new pair µ k (B, X) = (B ′ , X ′ ), where B ′ = µ k (B) and In this case, (X k , X ′ k ) is called an X -exchange pair.Definition 2.6 (A -seed mutation and A -exchange pair).Let (B, X, A) be an Aseed over P. Define the mutation of (B, X, A) at k ∈ {1, . . ., n} as a new triple

In this case, (A
is called the k-th exchange binomial of (B, X, A).
It is not hard to check that each mutation µ k maps a seed (X -seed or cluster Poisson seed or A -seed) to a new seed of the same type and that µ k is an involution.
Let T n be the n-regular tree.Let us label the edges of T n by 1, . . ., n such that the n different edges adjacent to the same vertex of T n receive different labels.
Definition 2.7 (Seed pattern).A seed pattern S over P is an assignment of a seed Σ t (X -seed or cluster Poisson seed or A -seed) to every vertex t of the n-regular tree T n such that Σ t ′ = µ k (Σ t ) for any edge t k t ′ .
We often fix a vertex t 0 ∈ T n as the rooted vertex of T n .For a seed pattern, the seed at the rooted vertex t 0 is called the initial seed.It is easy to see that a seed pattern is completely determined by its initial seed.Now we give some symbols which are used in the sequel.We always write B t = (b t ij ), X t = (X 1;t , . . ., X n;t ) and A t = (A 1;t , . . ., A n;t ).For simplicity, we will also use t to denote the seed at t ∈ T n .
Two (labeled) seeds are equivalent if they are the same up to relabeling.
Definition 2.8 (Exchange graph).Let S be a seed pattern.The exchange graph EG(S) of S is a graph whose vertices are in bijection with the seeds (up to equivalence) of S and whose edges correspond to the seed mutations.
Proposition 2.9.([FZ07, Proposition 3.9]).Let S be an A -seed pattern.For each A -seed t = (B t , X t , A t ), let X t = ( X 1;t , . . ., X n;t ) be the n-tuple of elements in F given by Then S = {(B t , X t )} t∈Tn forms an X -seed pattern.
Now we recall some seed patterns (X -seed patterns or A -seed patterns) over particular semifields.
• A seed pattern S over P is said to be of geometric type, if P is a tropical semifield, say P = Trop(Z 1 , . . ., Z m ).• A seed pattern S over P is said to be with principal coefficients at t 0 , if P = Trop(Z 1 , . . ., Z n ) and X i;t0 = Z i for i = 1, . . ., n. • A seed pattern S over P is said to be with universal coefficient semifield, if P = Q sf (X 1;t0 , . . ., X n;t0 ) for some seed t 0 of S.
Definition 2.10 (Coefficient matrices and extended exchange matrices).Let S be a seed pattern of geometric type with coefficient semifield P = Trop(Z 1 , . . ., Z m ).
We know that each X -variable X k;t has the form Proposition 2.11.([FZ02, Proposition 5.8]).Let S be a seed pattern of geometric type.Then for any edge , where B t and B t ′ are the extended exchange matrices at t and t ′ .
Definition 2.12 (Cluster Poisson algebra).Let S uc be an X -seed pattern with universal coefficient semifield.The cluster Poisson algebra X = X (S uc ) associated with S uc is the intersection where L(t) = K[X ±1 1;t , . . ., X ±1 n;t ].Definition 2.13 (Cluster algebra and upper cluster algebra).Let S be an A -seed pattern over a semifield P.
(i) The cluster algebra A = A(S) associated with S is the KP-subalgebra of F generated by the cluster variables of S, namely, (ii) The upper cluster algebra U = U(S) associated with S is the intersection where . Since cluster Poisson algebras and (upper) cluster algebras are defined from seed patterns, we can talk about the exchange graphs of these algebras.We can also talk about the (upper) cluster algebras of geometric type, with principal coefficients and with universal coefficient semifield.
Theorem 2.14 (Laurent phenomenon and positivity).Let A be a cluster algebra with coefficient semifield P and initial seed t 0 .The following statements hold.
(i) ( [FZ02]).Each cluster variable A k;t of A can be written as a Laurent polynomial in ).The coefficients of the above Laurent polynomial are in NP.

(iii) ([FZ03]
).If P = Trop(Z 1 , . . ., Z m ), then each cluster variable A k;t of A can be written as a Laurent polynomial in Z[Z 1 , . . ., Z m ][A ±1 1;t0 , . . ., A ±1 n;t0 ].A geometric upper cluster algebra with initial seed t 0 is called a full rank upper cluster algebra if its initial extended exchange matrix B t0 has full rank.
Remark 2.15.For a full rank upper cluster algebra U, we know that every extended exchange matrix B t of U has full rank, by Proposition 2.2.
Theorem 2.16.(Starfish Theorem, [BFZ05, Corollary 1.9]).Let U be a full rank upper cluster algebra and t 0 a seed of U. Then we have where t j = µ j (t 0 ) for j = 1, . . ., n and Theorem 2.17.([CL20, Theorem 10]).Let A be a cluster algebra with coefficient semifield P. Then the seeds of A whose clusters contain particular cluster variables form a connected subgraph of the exchange graph of A.
The following result is a direct corollary.
Corollary 2.18.([CL20, Corollary 3]).Let A be a cluster algebra with coefficient semifield P and t 1 , t 2 two seeds of A. If t 1 and t 2 have at least n − 1 common cluster variables, then in the exchange graph of A either t 1 and t 2 represent the same vertex or there is an edge between t 1 and t 2 .

d-vectors, f -vectors, g-vectors, c-vectors and
, where P M ∈ KP[A 1;t0 , . . ., A n;t0 ] with A j;t0 ∤ P M for j = 1, . . ., n.The vector is called the d-vector of M with respect to A t0 = (A 1;t0 , . . ., A n;t0 ) and the polynomial P M is called the numerator polynomial of M with respect to A t0 .
Remark 2.20.Notice that for any 0 = M, N ∈ L(t 0 ), we have Definition 2.21 (F -polynomial).Let B be an n × n skew-symmetrizable integer matrix and A a principal coefficient cluster algebra at t 0 with B t0 = B. Let A k;t be a cluster variable of A, which can be written as a Laurent polynomial in by the Laurent phenomenon.The polynomial Proposition 2.25.Let U be an upper cluster algebra and M a non-zero element in U. Let t 0 , t 1 be two seeds of is the (i, j)-entry of the D-matrix D (1) They are well-defined.
(2) The following statements are equivalent.
(i) There exists a cluster A t ′ containing both A i;t0 and A j;t ; (3) The following statements are equivalent.
(i) There exists no cluster A t ′ containing both A i;t0 and A j;t ; The following statements are equivalent. (i) The following statements are equivalent.
(i) A i;t0 = A j;t and there exists a cluster A t ′ containing both A i;t0 and Remark 2.31.(i) Roughly speaking, the integer (A i;t0 || A j;t ) d is defined to be the i-th component of the d-vector of A j;t with respect to A t0 .By Proposition 2.25 and Theorem 2.17, the d-compatibility degree (− || −) d is actually a well-defined function on A × (U\{0}), where U is the corresponding upper cluster algebra.
(ii) By Remark 2.20, for any 0 = M, N ∈ U, we have Recall that a cluster monomial in a seed t is a monomial in the cluster variables from t.
Corollary 2.32.Let A be a cluster algebra with coefficient semifield P and A k;t0 a cluster variable of A. Let t be a seed of A and So there must exist some j 0 ∈ I such that (A k;t0 || A j0;t ) d < 0. Then by Proposition 2.30 (4), we get A k;t0 = A j0;t .
(ii) Assume by contradiction there exists some i 0 ∈ I such that Then by Let A be a cluster algebra and A the set of cluster variables of A. We say that two cluster variables A i;t0 and A j;t of A are compatible if there exists a cluster A t ′ of A containing both A i;t0 and A j;t , which is equivalent to (A i;t0 || A j;t ) d ≤ 0, by Proposition 2.30 (2).A subset U of A is called a compatible set if any two cluster variables in U are compatible.
Theorem 2.33.([CL20, Theorem 13]).Let A be a cluster algebra and A the set of cluster variables of A. Then (i) a subset U of A is a compatible set if and only if it is a subset of some cluster of A; (ii) a subset U of A is a maximal compatible set if and only it is a cluster of A.

Unique factorization domains.
Recall that an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero.From now on, we always assume that R is an integral domain and we denote by R × the set of invertible elements in R.
Two non-zero elements r 1 , r 2 ∈ R are associate if there exists some invertible element s ∈ R × such that r 2 = sr 1 .A non-zero, non-invertible element r in R is irreducible, if any factorization r = r 1 r 2 with r 1 , r 2 ∈ R implies that either r 1 or r 2 belongs to R × .A non-zero, non-invertible element r in R is prime, if whenever r | r 1 r 2 for some r 1 , r 2 ∈ R, then r | r 1 or r | r 2 .Every prime element is irreducible, but the converse is not true in general.
Definition 2.34.An integral domain R is factorial if the following hold: (i) Every non-zero, non-invertible element r ∈ R can be written as a product for all i and j, then s = t and there is a bijection σ : {1, . . ., s} → {1, . . ., s} such that a i and b σ(i) are associate for all 1 ≤ i ≤ s.
Factorial domains are also known as unique factorization domains.
It is easy to see that in a factorial domain, all irreducible elements are prime.
Theorem 2.35.Let A be a geometric cluster algebra with coefficient semifield P = Trop(Z 1 , . . ., Z m ) and U the corresponding upper cluster algebra.Then the following statements hold.
(ii) The proof is same as that of (i) given in [GLS13].

The valuation pairing on an upper cluster algebra
In this section, we introduce the valuation pairing (− || −) v on an upper cluster algebra and prove the local unique factorization property for full rank upper cluster algebras.
Definition 3.1 (Valuation pairing).Let U be an upper cluster algebra and A the set of cluster variables of U. The pairing Example 3.2.Let U be an upper cluster algebra with a seed t and M a cluster variable in t.Then Now we summarize some useful and easy facts on the valuation pairing in the following proposition.
Proposition 3.3.Let U be an upper cluster algebra and t 0 a seed of U. Let M and L be two elements in U.The following statements hold.
In particular, Without loss of generality, we can assume that k = n.As an element in where L s is a Laurent polynomial in KP[A ±1 1;t0 , . . ., A ±1 n−1;t0 ], i.e., L s does not contain any A ±1 n;t0 .By M = 0, we know that there exists some s 0 such that L s0 = 0. Let t 1 = µ n (t 0 ) and thus t 0 = µ n (t 1 ).Applying the exchange relation, we have 1;t1 , . . ., A ±1 n−1;t1 ], we know that there must exist some m 0 large enough such that Set m = m 0 + s 0 .Then we know that (iii) This follows from (ii) and the fact ( The results in (iv) and (v) follow from the definition of the valuation pairing.
Definition 3.4 (Local factorization).Let U be an upper cluster algebra and t a seed of U.
Proposition 3.5 (Existence of local factorization).Let U be an upper cluster algebra with coefficient semifield P and t 0 a seed of U. Then any 0 = M ∈ U admits a local factorization with respect to t 0 .
Proof.We first define a cluster monomial N = A m1 1;t0 • • • A mn n;t0 in t 0 .Let m 1 := (A 1;t0 || M ) v and m 2 , . . ., m n are defined by induction.If m k has been defined, m k+1 is defined as follows: Thus the non-negative integers m 1 , . . ., m n are defined.So we get a cluster monomial N = A m1 1;t0 • • • A mn n;t0 in t 0 and we know that L := M/N ∈ U. Now we show that L = M/N satisfies (A k;t0 || L) v = 0, that is, L/A k;t0 / ∈ U for k = 1, . . ., n. Assume by contradiction that there exists some k ∈ {1, . . ., n} such that Thus we get which contradicts the choice of m k .So L/A k;t0 / ∈ U for k = 1, . . ., n.Thus M = N • L is a local factorization of M with respect to t 0 .This completes the proof.
Lemma 3.6 (Reduction Lemma).Let U be a full rank upper cluster algebra and t 0 a seed of U. Let L be a non-zero element in U and N = A m1 1;t0 • • • A mn n;t0 a cluster monomial in t 0 .Then the following statements hold.
Proof.(i) By the Starfish Theorem 2.16, we know that U = n i=0 L(t i ).Since A k;t0 is invertible in L(t i ) for any i ∈ {0, 1, . . ., n}\{k}, we have L/A s k;t0 ∈ L(t i ) for any i ∈ {0, 1, . . ., n}\{k}.Thus we get that L/A s k;t0 ∈ U if and only if L/A s k;t0 ∈ L(t k ).(ii) This follows from (i) and the definition of the valuation pairing.(iii) Without loss of generality, we assume k = 1.Since A m2 2;t0 • • • A mn n;t0 is invertible in L(t 1 ) and by (ii), we have Then by Proposition 3.3 (v), we get This completes the proof.
Theorem 3.7 (Local unique factorization property).Let U be a full rank upper cluster algebra and t 0 a seed of U. Then any 0 = M ∈ U admits a unique local factorization with respect to t 0 .
Proof.The existence of local factorization of M with respect to t 0 is known from Proposition 3.5.Now we show the uniqueness.Let M = N • L be a local factorization of M with respect to t 0 , where N = A m1 1;t0 • • • A mn n;t0 is a cluster monomial in t 0 and L ∈ U satisfies (A k;t0 || L) v = 0 for k = 1, . . ., n.By the Reduction Lemma 3.6 (iii), we have So m k = (A k;t0 || M ) v is uniquely determined by M .Namely, N is uniquely determined by M .This completes the proof.Now we give a counter-example to the local unique factorization property in the case where B t0 is not full rank.
Example 3.8.Let U be a geometric upper cluster algebra with initial extended exchange matrix B t0 given by Clearly, B t0 is not full rank.Denote t 1 = µ 1 (t 0 ) and t 3 = µ 3 (t 0 ).Applying the exchange relations, we have the following equality.
It is easy to check that both A 1;t1 and A 3;t3 are not divisible by any cluster variable in t 0 (alternatively, one can also refer to Theorem 5.4 (ii)).So the equality (3.1) gives two different local factorizations of A 2;t0 + 1 with respect to the initial seed t 0 .

Application to factoriality of upper cluster algebras
In this section we give several equivalent characterizations for the factoriality of upper cluster algebras.As an application, we show that full rank, primitive upper cluster algebras are factorial.4.1.Factoriality of partially compactified upper cluster algebras.In this subsection, we reduce the factoriality of partially compactified upper cluster algebras to the factoriality of upper cluster algebras.
Given a geometric upper cluster algebra U, let A be the corresponding cluster algebra.By definition, all the frozen variables are invertible in U and A. However, when studying the cluster structure on the coordinate rings of various algebraic varieties, it is important to allow that some of the frozen variables are not inverted.
We use inv to denote a subset of the set The partially compactified upper cluster algebra U(inv) is defined to be The partially compactified cluster algebra A(inv) is defined to be . Clearly, we have the inclusions U(inv) ⊆ U and A(inv) ⊆ A. Moreover, the equality A domain R is said to be atomic if every non-zero, non-invertible element r ∈ R can be decomposed as a product of irreducible elements.Remark 4.2.In the above lemma, the assumption that R is atomic is missing in the original statement of [GELS19, Corollary 1.20].We thank the first and third authors of [GELS19] for pointing this out in an email message [GES22].In the same email message, they also included a proof that any geometric (upper) cluster algebra is an atomic domain.Their proof actually also works for partially compactified (upper) cluster algebras.
The proof of the above proposition is put in the appendix.We remark that the main idea for the proof follows that of [GES22] with some modifications and simplifications.
Lemma 4.4.Let M be an element in U. Then M belongs to U(inv) if and only if M belongs to L(t, inv) for some seed t.
"⇐=": Assume that M ∈ U belongs to L(t, inv) for some seed t.Using similar arguments as in the proof of Proposition 2.25, we get M ∈ L(µ k (t), inv), where k = 1, . . ., n.Then by induction, we have M ∈ L(t ′ , inv) for any seed t ′ of U(inv).Hence, we have M ∈ U(inv).Thanks to the above proposition, we will mainly focus on the study the factoriality of U.

4.2.
Characterizations for the factoriality of upper cluster algebras.We first give an observation which shows why the valuation pairing can be used to study the factoriality of upper cluster algebras.
Proposition 4.7 (Observation).Let U be a geometric upper cluster algebra and A k;t a cluster variable of U. Then the following two statements are equivalent.
(i) A k;t is prime in U.
(ii) For any non-zero elements M and L in U, we have the following equality: Proof.(i)⇒ (ii): This is clear from the definition of the valuation pairing.
(ii)⇒ (i): Otherwise, we have that both M and L are not divisible by A k;t in U. Then we have

Now by our assumption, we have
By Theorem 2.35 (ii), we know that A k;t is not invertible in U. Hence, A k;t is prime in U.
It is natural to ask under which conditions the equality (4.1) always holds.Thanks to Reduction Lemma 3.6, we can answer this question for full rank upper cluster algebras.
Proposition 4.8.Let U be a full rank upper cluster algebra and t a seed of U. Put t k = µ k (t) and let P k;t be the k-th exchange binomial of t.Then the following statements are equivalent.
Proof.(i)=⇒ (ii): By our assumption in Subsection 1.4, we know that A k;t is not invertible in L(t k ).Now assume by contradiction that A k;t is not prime in L(t k ).Then there exists M ′ , L ′ ∈ L(t k ) such that both M ′ and L ′ are not divisible by Because A v t k is invertible in L(t k ), we know that both M and L are not divisible by A k;t in L(t k ), but the product M • L is divisible by A k;t in L(t k ).By the Reduction Lemma 3.6 (ii), we have (iii)⇐⇒ (iv and it is not divisible by any Z i and A j;t k , we know that (iv)⇐⇒ (v): This follows from the fact that the polynomial ring Theorem 4.9.Let U be a full rank upper cluster algebra with initial seed t 0 .Then the following statements are equivalent.
for any non-zero elements M and L in U. (iv) Any exchange binomial P k;t0 of t 0 is irreducible in Proof.(i)=⇒ (ii): By Theorem 2.35 (ii), we know that any cluster variable is irreducible in U.Because U is factorial, we get that any irreducible element is prime in U.In particular, any initial cluster variable A k;t0 is prime in U.
(ii)=⇒ (i): Let S be the multiplicative set generated by initial cluster variables.By the Laurent phenomenon, we have 1;t0 , . . ., A ±1 n;t0 ], which is factorial.Since the initial cluster variables are prime in U and by Lemma 4.1 and Proposition 4.3, we get that U is factorial.
Remark 4.10.Garcia Elsener et al. in [GELS19, Theorem 5.1] prove the equivalence of (i) and (iv) in Theorem 4.9 for geometric cluster algebra A (possibly not full rank) with an acyclic initial seed.Note that in acyclic case, A = U, by Proposition 2.19.
Our method here is very different with that in [GELS19].
4.3.Full rank, primitive upper cluster algebras are factorial.In this subsection, we show that full rank, primitive upper cluster algebras are factorial.
Recall that a vector (b Definition 4.11 (Primitive upper cluster algebra).Let U be a geometric upper cluster algebra with initial seed t 0 .We say that U is a primitive upper cluster algebra, if the initial extended exchange matrix B t0 is primitive.Proposition 4.12.Let U be a primitive upper cluster algebra.Then for any seed t, the following statements hold.
where k = 1, . . ., n. Theorem 4.13.Let U be a full rank, primitive upper cluster algebra with initial seed t 0 , and let be a non-initial cluster variable of U, where Proof.(i) By Proposition 4.12 (ii), we know that every exchange binomial of t 0 is irreducible in the corresponding polynomial ring.Then the result follows from Theorem 4.9 (i)(iv).
(ii) We first show that P M is not invertible in K[Z 1 , . . ., Z m ][A 1;t0 , . . ., A n;t0 ].Since M is a non-initial cluster variable and by Proposition 2.30 (4), we have d ∈ N n and thus A d t0 ∈ U.By M = P M /A d t0 , we get M A d t0 = P M .By Theorem 2.35 (2), we know that both M and A d t0 are not invertible in U.So P M = M A d t0 is not invertible in U. Namely, we have ].We claim that P M is not divisible by any Z i and A j;t0 in Since P M is a numerator polynomial, it is not divisible by any A j;t0 .There are two approaches to prove Z i ∤ P M .The first one is that we prove the result by induction on the minimal length of a sequence of mutations from the initial seed to the final seed.The second one is that we just view Z i as an exchangeable cluster variable.Then by the fact that M and Z i are in the same cluster and they are not equal, we get (Z i || M ) d = 0, by Proposition 2.30 (5).This implies that P M is not divisible by Z i .
We claim that P M is not invertible in ].This follows from the two facts that P M is not invertible in , we get which is an equality in U, since d, v 1 , v 2 ∈ N n .Because U is factorial and M is irreducible in U, we get that either U 1 or U 2 is divisible by M in U. Without loss of generality, we assume that U ′ 1 := U 1 /M ∈ U. Then by the equality (4.2), we get , which implies that both U ′ 1 and U 2 are invertible in L(t 0 ).So L 2 = U 2 /A v2 t0 is invertible in L(t 0 ).Hence, P M is irreducible in L(t 0 ).Since P M is irreducible in L(t 0 ) and by the fact that P M is not divisible by any Z i and ].This completes the proof.
Remark 4.14.(i) Note that any principal coefficient upper cluster algebra is primitive and has full rank.So principal coefficient upper cluster algebras are always factorial.
(ii) A different proof of Theorem 4.13 (i) via the factoriality of Cox rings can be found in [GHK15, Corollary 4.7].
(iii) Garcia Elsener et al. in [GELS19, Corollary 5.3] prove that any principal coefficient cluster algebra A with an acyclic initial exchange matrix is factorial.Note that in the acyclic case, we have A = U, by Proposition 2.19.
Corollary 4.15.Let U be a full rank, primitive upper cluster algebra and A k;t0 a cluster variable of U. Then for any non-zero elements M and L in U, we have Proof.This follows from Theorem 4.9 (i)(iii) and Theorem 4.13 (i).

4.4.
Examples of non-factorial upper cluster algebras.Proposition 4.16.([GLS13, Corollary 6.5]).Let A be a geometric cluster algebra with initial extended exchange matrix where c ≥ 1 and d ≥ 3 an odd number.Then A is not factorial.
Remark 4.17.Note that in Proposition 4.16, A is an acyclic cluster algebra.In this case, A coincides with its upper cluster algebra U, by Proposition 2.19.So Proposition 4.16 provides many upper cluster algebras which have full rank but are not factorial.
Let U be the geometric upper cluster algebra with initial extended exchange matrix We know that U is a full rank cluster algebra.It is non-factorial, by Remark 4.17.
The following is a concrete example where U is primitive but not factorial.
Example 4.18.Let A be a geometric cluster algebra with the following initial extended exchange matrix It is easy to see that B t0 is primitive and acyclic but not full rank.It is known from [GLS13, Proposition 6.1] that A is not factorial.Since A is acyclic, A coincides with its upper cluster algebra U, by Proposition 2.19.So U = A is not factorial.
4.5.Ray Fish Theorem.Recall that the Starfish Theorem states that any full rank upper cluster algebra U can be written as the intersection of n + 1 Laurent polynomial rings.In this subsection, we show that any full rank, primitive upper cluster algebra can be written as the intersection of two Laurent polynomial rings.We call this the Ray Fish Theorem.
Corollary 4.21.Let U be a full rank, primitive upper cluster algebra and A the corresponding cluster algebra.Then A = U if and only if A is factorial.
Proof."=⇒": Since U is full rank and primitive and by Theorem 4.13, we know that U is factorial.Then the factoriality of A follows from A = U.
Lemma 4.22.Let U be a geometric upper cluster algebra with coefficient semifield P = Trop(Z 1 , . . ., Z m ) and A i;t0 , A j;t two cluster variables of U. Then A i;t0 and A j;t are associate in U if and only if A i;t0 = A j;t .
Proof.Assume that A i;t0 and A j;t are associate in U. Then for some λ ∈ K × and c 1 , . . ., c m ∈ Z, by Theorem 2.35 (ii).Notice that the equality (4.3) can be viewed as the Laurent expansion of A j;t with respect to t 0 .It is easy to see that (A i;t0 || A j;t ) d = −1.Then by Proposition 2.30 (4), we get A j;t = A i;t0 .
Conversely, assume A i;t0 = A j;t .Clearly, we have that A i;t0 and A j;t are associate in U.This completes the proof.
Inspired by Proposition 4.19, we give the following result.
Since the inclusion U ⊆ L(t) ∩ L(t 0 ) is clear, it suffices to show the converse inclusion.We know that any M ∈ L(t) ∩ L(t 0 ) has the following form: So we have L(t) ∩ L(t 0 ) ⊆ U and thus U = L(t) ∩ L(t 0 ).

Applications to d-vectors
5.1.d-compatibility degree and d-vectors via the valuation pairing.In this subsection, we show how to express the d-compatibility and the d-vectors using the valuation pairing for full rank upper cluster algebras.
Theorem 5.1.Let U be a full rank upper cluster algebra with initial seed t 0 .Let be a non-zero element in U, where d M := (d 1 , . . ., d n ) T is the d-vector of M with respect to A t0 .Then the following statements hold.
(i) For any k = 1, . . ., n, we have Proof.(i) We know that there exist two vectors By the Reduction Lemma 3.6 (iii), we get The results in (ii) and (iii) follow from (i).

Local factorizations of cluster monomials.
Let U be an upper cluster algebra with coefficient semifield P and initial seed t 0 .In this subsection, we prove that if M is a cluster monomial in non-initial cluster variables, then M/A k;t0 / ∈ U, that is, (A k;t0 || M ) v = 0 for k = 1, . . ., n.In particular, we can give a local factorization for any cluster monomial.Lemma 5.2.Let U be an upper cluster algebra with coefficient semifield P and t a seed of U.
Proof.Assume by contradiction that there exists some b i0 < 0. Without loss of generality, we just assume b 1 < 0. Let t ′ = µ 1 (t) and thus t = µ 1 (t ′ ).Applying the exchange relation, we know that where P is the first exchange binomial of t ′ .Then we know that which is the expansion of N with respect to A t ′ .By N ∈ U, we get that Thus we have 1/P −b1 ∈ L(t ′ ).Because both P −b1 and 1/P −b1 are in L(t ′ ), we must have P −b1 = λA c1 1;t ′ • • • A cn n;t ′ for some invertible element λ in KP and c 1 , . . ., c n ∈ Z. Since P is a polynomial in KP[A 2;t ′ , . . ., A n;t ′ ] with A i;t ′ ∤ P for any i, we get that c 1 = . . .= c n = 0 and P −b1 = λ is invertible in KP.By our assumption in Subsection 1.4, we know that P −b1 = λ is not invertible in KP.This concludes a contradiction.So we must have b 1 , . . ., b n ≥ 0.
Proposition 5.3.Let U be an upper cluster algebra with coefficient semifield P and t 0 , t two seeds of U.
If there exists an invertible element λ in KP such that M = λN , then (i) N is a cluster monomial in A t , i.e., we have b 1 , . . ., b n ≥ 0; (ii) there exists a bijection σ : I → J such that A i;t0 = A σ(i);t and c i = b σ(i) .
In particular, we have M = N .
Proof.(i) By M = λN , we know that N = λ −1 M ∈ U. Then by Lemma 5.2, we know that N is a cluster monomial in A t .
(ii) Let A j;t be a cluster variable appearing in N .Then by (i), we have b j > 0 and j ∈ J.Because there must exist some j ′ ∈ I such that (A j;t || A j ′ ;t0 ) d < 0.
Then by Proposition 2.30 (4), we have A j;t = A j ′ ;t0 .So we have Similarly, we can show that any cluster variable A i;t0 appearing in M also appears in N , and the multiplicity of A i;t0 in N is equal to the multiplicity of A i;t0 in M .Then the result follows.
Theorem 5.4 (Local factorization of cluster monomials).Let U be an upper cluster algebra with coefficient semifield P and initial seed t 0 .Then the following statements hold.
(i) If L is a cluster monomial of U which does not contain the initial cluster variable A k;t0 for some k, then A bi i;t be a cluster monomial in a seed t of U and Proof.(i) Assume by contradiction that L/A k;t0 ∈ U.In the following proof, we will deduce a contradiction.be a monomial in cluster variables of U (not necessarily a cluster monomial), where d M := (d 1 , . . ., d n ) T is the d-vector of M with respect to A t0 .Then the following statements hold.
(i) If M does not contain the initial cluster variable A k;t0 for some k, then we have Since U is a full rank, primitive upper cluster algebra, we know that U is factorial, by Theorem 4.13.Then by Theorem 4.9 (i)(iii), we know that Then by Theorem 5.1 (ii), we get (ii) This follows from (i).

Application to F -polynomials
In this section, we always assume that U is an upper cluster algebra with principal coefficients at t 0 .6.1.From F -polynomials to monomials in non-initial cluster variables.In this subsection, we prove that if M is a monomial in non-initial cluster variables, then M is uniquely determined by its F -polynomial.
Let M be a monomial in cluster variables of U, say M = s i=1 M i , where each M i is a cluster variable of U. The F -polynomial F M of M is defined to be the polynomial F M := s i=1 F Mi , where F Mi is the F -polynomial of the cluster variable M i .The g-vector of M is defined to be the vector g M := s i=1 g Mi , where g Mi is the g-vector of M i .Theorem 6.1.Let U be an upper cluster algebra with principal coefficients at t 0 .Let M, N be two monomials in non-initial cluster variables of U and F M , F N their F -polynomials.If F M = F N , then M = N .In particular, g M = g N , where g M and g N are g-vectors of M and N .
Proof.Let d M and d N be the d-vectors of M and N with respect to A t0 .We know that M and N have the form M = P M /A dM t0 and N = P N /A dN t0 , (6.1) where P M , P N ∈ Z[Z 1 , . . ., Z n ; A 1;t0 , . . ., A n;t0 ] with A i;t0 ∤ P M and A i;t0 ∤ P N for any i = 1, . . ., n.
Because P M and P N are not divisible by any A i;t0 , we know that d M + g M and d N + g N are the d-vectors of F M ( X 1;t0 , . . ., X n;t0 ) and F N ( X 1;t0 , . . ., X n;t0 ) with respect to t 0 .By F M = F N , we have F M ( X 1;t0 , . . ., X n;t0 ) = F N ( X 1;t0 , . . ., X n;t0 ), which implies that d M + g M = d N + g N and P M = P N .
By P M = P N and applying Proposition 5.6 to M and N , we get d M = d N .Then by the equality (6.1), we know that M = N .In particular, we have g M = g N .This completes the proof.6.2.F -polynomials of non-initial cluster variables are irreducible.In this subsection, we prove that the F -polynomials of non-initial cluster variables are irreducible in K[Z 1 , . . ., Z n ].Theorem 6.2.Let U be an upper cluster algebra with principal coefficients at t 0 and A k;t a non-initial cluster variable of U. Then the F -polynomial F k;t of A k;t is irreducible in K[Z 1 , . . ., Z n ].
Proof.We claim that F k;t is not a constant.Otherwise, the f -compatibility degree (A i;t0 || A k;t ) f = 0 for i = 1, . . ., n.Then by Proposition 2.30 (2), we get that A k;t is compatible with any cluster variable in A t0 .Thus A k;t ∈ A t0 , by Theorem 2.33.This contradicts that A k;t is a non-initial cluster variable of U.So F k;t is not a constant.
Thus we have A k;t = A v t0 (L 1 L 2 ), where v = g + v 1 + v 2 − d 1 − d 2 .Now we show that v = 0 and A k;t = L 1 L 2 .We choose two vectors v + and v − in N n such that v = v + − v − .Then by we get that Since F k;t is not a constant, we know that A k;t is not an initial cluster variable.Then by Theorem 5.4, we have (A j;t0 || A k;t ) v = 0 for j = 1, . . ., n.On the other hand, we know that L 1 L 2 also satisfies (A j;t0 || L 1 L 2 ) v = 0 for j = 1, . . ., n.So the equality (6.3) gives two local factorizations of M with respect to t 0 .Then by the uniqueness in Theorem 3.7, we get that Remark 6.3.Garcia Elsener et al. in [GELS19, Theorem 3.9] prove that the Fpolynomials of non-initial cluster variables are irreducible for factorial principal coefficient cluster algebras.In fact, their proof still works for factorial principal coefficient upper cluster algebras.Note that the factoriality of principal coefficient upper cluster algebras is no longer a problem, thanks to Theorem 4.13.Thus one can also use the method in [GELS19] to show the irreducibility of F -polynomials of non-initial cluster variables.
Since each a i is not invertible in R and, by the assumption, we know that it is also not invertible in U(inv).Then by Lemma 7.13 (i), we have λ(a i ) ∈ Z ≥1 .Hence, λ(r) = λ(a 1 ) + . . .+ λ(a s ) ≥ s.
Since s is bounded by λ(r), there exists a decomposition r = a 1 • • • a s such that s is maximal.We claim that in such a decomposition, each a i is irreducible in R. Otherwise, there exists some i 0 such that a i0 is not irreducible in R. Without loss of generality, we can assume i 0 = s.Then there exist two non-invertible elements a ′ s , a ′′ s in R \ {0} such that a s = a ′ s a ′′ s .Thus we have r = a 1 • • • a s−1 a ′ s a ′′ s , which is a decomposition of r of length s + 1 satisfying that each factor is not invertible in R \ {0}.This contradicts the choice of s.Hence, r = a 1 • • • a s is a decomposition of r ∈ R satisfying that each a i is irreducible in R. Therefore, R is atomic.
Proof of Proposition 4.3.Thanks to Lemma 7.12, we can apply Lemma 7.14 to the case R = U(inv) or R = A(inv).Hence, U(inv) and A(inv) are atomic.
Lemma 4.1.([GELS19, Corollary 1.20]).Let R be an atomic domain and S ⊆ R \ {0} a multiplicative set generated by prime elements.Then R is factorial if and only if the localization S −1 R is.
Proposition 4.6.U(inv) is factorial if and only if U is factorial.Proof.By Lemma 4.5, we know that the non-invertible frozen variables Z 1 , . . ., Z p are prime elements in U(inv).Let S be the multiplicative set in U(inv) generated by the prime elements Z 1 , • • • , Z p .Clearly, we have the inclusion S −1 (U(inv)) ⊆ U. Using Lemma 4.4, one can easily show the converse inclusion.Hence, we have S −1 (U(inv)) = U.Then the result follows from Proposition 4.3 and Lemma 4.1.
Basics on cluster algebras and cluster Poisson algebras.An n × n integer matrix B is said to be skew-symmetrizable if there is an integer diagonal matrix S with strictly positive diagonal entries such that SB is skew-symmetric.Such an S is said to be a skew-symmetrizer of B.
Z n ], which only depends on (B, t 0 ; k, t), is called an F -polynomial of B. Sometimes we view F B;t0 k;t as a polynomial in new variables Y 1 , . .., Y n due to the convention in [FZ07].Proposition 2.22.([FZ07,Proposition 5.2]).The F -polynomial F B;t0 k;t (Y 1 , . .., Y n ) is not divisible by any Y j .Let B be an n × n skew-symmetrizable matrix and t 0 a vertex of T n .Now we define two families of integer matrices {D B;t0 Definition 2.23 (D-matrices and F -matrices).Let B be an n×n skew-symmetrizable matrix and t 0 a vertex of T n .
lk }, if j = k, for any edge t k t ′ in T n .The matrices {D B;t0 t } t∈Tn are called the D-matrices of B. (ii) ([FG19, Definition 2.6]).Let F B;t0 k;t (Z 1 , . . ., Z n ) be the F -polynomial given in Definition 2.21, and f B;t0 ik;t be the maximal exponent of Z i appearing in F B;t0 k;t .n×n is called an F -matrix of B. Remark 2.24.Let A be a cluster algebra with initial exchange matrix B at t 0 .Then the k-th column vector d B;t0 k;t of D B;t0 t is exactly the d-vector d t0 (A k;t ) of A k;t with respect to A t0 , by [FZ07, (7.7)].
the d-vectors of M with respect to t 0 and t 1 respectively.Thend i = d ′ i for any i = k.Proof.The proof is the same as that of [RS18, Proposition 2.5].Theorem 2.28.([NZ12, (3.11)], [CL18, Corollary 4.11]).Let B be an n × n skew-symmetrizable matrix and S a skew-symmetrizer of B. Then for any vertices t 0 , t ∈ T n , we have SC B;t0 [FG23].Then for any cluster variable A k;t of A, there exists a unique vector g B;t0 k;t = (g B;t0 1k;t , ..., g B;t0 nk;t ) T ∈ Z n such thatA k;t = A g B;t 0 k;t t0 • F B;t0 k;t ( X 1;t0 , ..., X n;t0 ), T ∈ Z n , which only depends on (B, t 0 ; k, t), is called a c-vector of B. The matrix C B;t0 t = (c B;t0 ij;t ) n×n is called a C-matrix of B at t. t S −1 (G B;t0 t ) T = I n .2.3.Compatibility degrees on the set of cluster variables.In this subsection, we recall that the d-compatibility degree given in[CL20]and f -compatibility degree given in[FG23].Definition 2.29 (Compatibility degrees).Let A be a cluster algebra and A the set of cluster variables of A. (i) ([CL20, Definition 8]).The d-compatibility degree (− || −) d : A × A → Z ≥−1 is defined by (A i;t0 || A j;t ) d := d there must exist some i 1 ∈ I such that (A k;t0 || A i1;t ) d < 0. So we get A k;t0 = A i1;t , by Proposition 2.30 (4).Thus A i0;t and A k;t0 = A i1;t are in the same cluster.Then by Proposition 2.30 (2), we get (A k;t0 || A i0;t ) d ≤ 0. This contradicts (A k;t0 || A i0;t ) d > 0. So (A k;t0 || A i;t ) d ≤ 0 for any i ∈ I.
for any non-zero elements M and L in U. Hence, A k;t is prime in L(t k ).(ii)=⇒ (i): This follows from the Reduction Lemma 3.6 (ii).(ii)⇐⇒ (iii): Applying the exchange relation, we have A k;t A k;t k = P k;t .Since A k;t k is invertible in L(t k ), we know that A k;t is prime in L(t k ) if and only if P k;t is prime in L(t k ).
KP[A 1;t , . . ., A n;t ], Q ∈ KP[A 1;t0 , . . ., A n;t0 ] and v i , s i ≥ 0 for any i = 1, . . ., n.Thus we have A s1 1;t0 By Theorem 2.35 (ii), we know that A i;t0 and A j;t are irreducible in U for any i and j.By Lemma 4.22 and the fact that t and t 0 have no common cluster variables, we know that A i;t0 and A j;t are non-associate for all 1 ≤ i, j ≤ n.Since U is full rank and primitive and by Theorem 4.13, we know that U is factorial.
a monomial in non-initial cluster variables, then d M and M are uniquely determined by the numerator polynomial P M .