  We study the Laurent property for autonomous and nonautonomous discrete
equations. First we show, without relying on the caterpillar lemma, the Laurent
property for the Hirota-Miwa and the discrete BKP equations. Next we introduce
the notion of reductions and gauge transformations for discrete bilinear
equations and we prove that these preserve the Laurent property. Using these
two techniques, we obtain the explicit condition on the coefficients of a
nonautonomous discrete bilinear equation for it to possess the Laurent
property. Finally we study the denominators of the iterates of an equation with
the Laurent property and we show that any reduction to a mapping on a
one-dimensional lattice of a nonautonomous Hirota-Miwa equation or discrete BKP
equation, with the Laurent property, has zero algebraic entropy.
