Laurent phenomenon

Remarks

The theorem is due to Fomin and Zelevinsky. Each mutation divides by a cluster variable, so a priori the cluster variables are only rational functions of the initial ones. The content of the theorem is that all these denominators are monomials. The proof goes by induction on the length of a mutation sequence, and it uses that the exchange polynomials of neighbouring mutations are coprime. Example: for $n = 2$ and $B = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$, the cluster variables satisfy $x_{m+1} x_{m-1} = x_m + 1$. Starting from $x_1, x_2$ one gets $x_3 = (x_2 + 1)/x_1$, $x_4 = (x_1 + x_2 + 1)/(x_1 x_2)$, $x_5 = (x_1 + 1)/x_2$, and then $x_6 = x_1$, $x_7 = x_2$. All five are Laurent polynomials in $x_1, x_2$. The special form of the exchange relations matters: a general rational recurrence produces denominators that are not monomials.

Going deeper

For a student

The Laurent phenomenon is one of the founding results of the theory of cluster algebras. Several nonlinear recurrences, such as the Somos-4 and Somos-5 recurrences, can be realized by mutations. So their terms are Laurent polynomials in the initial values, which explains the integrality of the Somos sequences for $k = 4, 5$. A stronger statement, that the coefficients of these Laurent polynomials are nonnegative, was conjectured by Fomin and Zelevinsky and later proved for skew-symmetric matrices $B$.