Let \dlap be the discrete Laplace operator acting on functions (or rational
matrices) f:QL→Q, where QL is the two
dimensional lattice of size L embedded in Z2. Consider a rational
L×L matrix H, whose inner entries Hij
satisfy \dlap\mathcal{H}_{ij}=0. The matrix H is thus the
classical finite difference five-points approximation of the Laplace operator
in two variables. We give a constructive proof that H is the
restriction to QL of a discrete harmonic polynomial in two
variables for any L>2. This result proves a conjecture formulated in the
context of deterministic fixed-energy sandpile models in statistical mechanics.Comment: 18 pag, submitted to "Note di Matematica