The Laurent Phenomenon

By Sergey Fomin, Andrei Zelevinsky. Advances in Applied Mathematics, 2001

362 citations recorded by the source corpus.

A composition of birational maps given by Laurent polynomials need not be given by Laurent polynomials; however, sometimes-quite unexpectedly-it does. We suggest a unified treatment of this phenomenon, which covers a large class of applications. In particular, we settle in the affirmative a conjecture of D. Gale and R. Robinson on integrality of generalized Somos sequences, and prove the Laurent property for several multidimensional recurrences, confirming conjectures by J. Propp, N. Elkies, and M. Kleber.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/3e411273911ad429d04687b3952f2e74b2fe3b9f

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Citations in the corpus

Canonical bases for cluster algebras (2014, background): “the dual basis and f′ i = d −1 i (e ′ i) ∗, a cluster monomial in this seed is a monomial on TN◦,w ⊂Aof the form zm with m= Pn i=1 aif ′ i and the ai non-negative for i∈Iuf. By the Laurent phenomenon [FZ02b], such a monomial always extends to a regular function on A. A cluster monomial on Ais a regu”
Positivity and canonical bases in rank 2 cluster algebras of finite and affine types (2003, background): “Since its inception, the theory of cluster algebras has developed several interesting connections: • Discrete dynamical systems [6].”
Recurrence Relations for Elliptic Sequences: Every Somos 4 is a Somos k (2004, methodology): “A different viewpoint on the apparent integrality of Somos sequences is given by the theory of cluster algebras, summarised by Fomin and Zelevinsky in [ 2 ].”
Moduli spaces of local systems and higher Teichmüller theory (2003, background): “This duality should also include the canonical bases of Lusztig and the Laurent phenomenon of Fomin and Zelevinsky [ FZ3 ] as special cases.”
Discrete monodromy, pentagrams, and the method of condensation (2007, background): “See [ FZ ], [Sp], and the references therein for work in this topic.”
Introduction to Cluster Algebras (2018, background): “The Laurent property originates from the study of several classical recurrences that seem as though they should produce rational numbers, but turn out to give only integers under certain initial conditions [15].”
Greedy elements in rank 2 cluster algebras (2012, background): “15), we see that this formula yields x[3, 3] = x 1 x −3 2 (”
Cluster Algebras and Poisson Geometry (2002, background): “mation rules for all the edges incident to one vertex of the tree, one can restore all the other transformation rules. The evolution of transformation rules provides the so-called Laurent phenomenon ([FZ3]), which means the following: ﬁx one cluster with cluster variables x1,...,xn and express any o”
2-frieze patterns and the cluster structure of the space of polygons (2010, background): “It follows from the Laurent phenomenon for cluster algebras [13] that all the entries of the frieze (5.”
Quantum toroidal algebras and their representations (2008, methodology): “Viewing the sequence inE, we get a solution which is polynomial in the monomials m with positive coecients; this is analogous to the Fomin{Zelevinsky Laurent Phenomena positivity properties for analogous systems (Y -systems) [ FZ1 , FZ2], as noticed in [He9].”
Framed BPS states (2010, neutral): “The Laurent phenomena of [38] allow us to construct formal laminations.”
Sigma function solution of the initial value problem for Somos 5 sequences (2005, background): “…to redistribution; see https://www.ams.org/journal-terms-of-use of the Laurent phenomenon: taking the initial data τ0, τ1, τ2, τ3 and the parameters α, β as variables, all subsequent terms τn for n ≥ 4 in the sequence are Laurent polynomials (rather than just rational functions) in these variables.”
Plane partitions in the work of Richard Stanley and his school (2015, background): “As Robbins writes in [83], after having in vain tried for some time to prove their 3It is interesting to note that this — at the time, isolated, and maybe somewhat obscure — object has now become part of a fascinating theory, namely the theory of discrete integral systems (cf. e.g. [23]), and is may”
Birational geometry of cluster algebras (2013, background): “ariety not obvious from the expression as a union of tori are easily seen from this construction. For example, it gives a simple geometric explanation for the Laurent phenomenon (originally proved in [FZ02b]), see Corollary 3.11. From the blowup picture it is clear that the Fock-Goncharov dual basis”
Y-systems and generalized associahedra (2001, neutral): “Direct check shows that in the case when h = 2e is even, the corresponding sequence of positive roots α(k) has the form α(1; 1), . . . , α(1;n), α(2; 1), . . . , α(2;n), . . . , α(e; 1), . . . , α(e;n).”