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Let $\dlap$ be the discrete Laplace operator acting on functions (or rationalmatrices) $f:\mathbf{Q}_L\to\mathbb{Q}$, where $\mathbf{Q}_L$ is the twodimensional lattice of size $L$ embedded in $\mathbb{Z}_2$. Consider a rational$L\times L$ matrix $\mathcal{H}$, whose inner entries $\mathcal{H}_{ij}$satisfy $\dlap\mathcal{H}_{ij}=0$. The matrix $\mathcal{H}$ is thus theclassical finite difference five-points approximation of the Laplace operatorin two variables. We give a constructive proof that $\mathcal{H}$ is therestriction to $\mathbf{Q}_L$ of a discrete harmonic polynomial in twovariables for any $L>2$. This result proves a conjecture formulated in thecontext of deterministic fixed-energy sandpile models in statistical mechanics.

On a class of rational matrices and interpolating polynomials related to the discrete Laplace operator