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

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dc.creatorVivo, Pierpaolo
dc.creatorCasartelli, Mario
dc.creatorDall'Asta, Luca
dc.creatorVezzani, Alessandro
dc.date2007-05-09
dc.date.accessioned2026-07-07T08:00:23Z
dc.date.available2026-07-07T08:00:23Z
dc.descriptionLet $\dlap$ be the discrete Laplace operator acting on functions (or rational matrices) $f:\mathbf{Q}_L\to\mathbb{Q}$, where $\mathbf{Q}_L$ is the two dimensional 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 the classical finite difference five-points approximation of the Laplace operator in two variables. We give a constructive proof that $\mathcal{H}$ is the restriction to $\mathbf{Q}_L$ 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.
dc.description18 pag, submitted to "Note di Matematica"
dc.identifierhttps://arxiv.org/abs/0705.1294
dc.identifierhttp://arxiv.org/abs/0705.1294
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/128656
dc.subjectMathematical Physics
dc.titleOn a class of rational matrices and interpolating polynomials related to the discrete Laplace operator
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