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

Abstract

Let be the discrete Laplace operator acting on functions(or rational matrices) ,where is the two dimensional lattice of size embedded in . Consider a rational matrix , whose inner entries satisfy . The matrix is thus theclassical finite difference five-points approximation of theLaplace operator in two variables. We give a constructive proofthat is the restriction to of adiscrete harmonic polynomial in two variables for any . Thisresult proves a conjecture formulated in the context ofdeterministic fixed-energy sandpile models in statisticalmechanics.