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


Let \dlap be the discrete Laplace operator acting on functions(or rational matrices) f:\mathbf{Q}_L\rightarrow\mathbb{Q},where \mathbf{Q}_L is the two dimensional lattice of size Lembedded 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 theLaplace operator in two variables. We give a constructive proofthat \mathcal{H} is the restriction to \mathbf{Q}_L of adiscrete harmonic polynomial in two variables for any L>2. Thisresult proves a conjecture formulated in the context ofdeterministic fixed-energy sandpile models in statisticalmechanics.

DOI Code: 10.1285/i15900932v28n2p1

Keywords: rational matrices; discrete Laplacian; discrete harmonic polynomials; sandpile

Classification: 11C99 (Polynomials and matrices)

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