Abstract: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 factor… Show more
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