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

Singularities of the discrete KdV equation and the Laurent property