Singularities of the discrete KdV equation and the Laurent property

We study the distribution of singularities for partial difference equations, in particular, the bilinear and nonlinear form of the discrete version of the Korteweg-de Vries (dKdV) equation. By the Laurent property, the irreducibility, and the co-primeness of the terms of the bilinear dKdV equation, we clarify the relationship of these properties with the appearance of zeros in the time evolution. The results are applied to the nonlinear dKdV equation and we formulate the famous integrability criterion (singularity confinement test) for nonlinear partial difference equations with respect to the co-primeness of the terms. (v2,v3,v4: minor revisions have been made)