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Solutions to Laplace's equation are called harmonic functions. Harmonic functions arise in many applications, such as physics and the theory of stochastic processes. Of interest classically are harmonic polynomials, which have a simple classification. Further, the work of Reznick, building on the work of others, namely Sylvester, Clifford, Rosanes, Gundelfinger, Cartan, Maass and Helgason, has led to a classification of all polynomial solutions to a differential equation of arising from a homogeneous polynomial over an algebraically closed field. The definition of harmonicity can be extended to the space of polynomials in free variables using the concept of a noncommutative Laplacian. Given a positive integer $\ell$, the $\ell$-Laplacian of a noncommutative (abbreviated NC) polynomial $p$ in the direction $h$ is defined to be $$ \lap_{\ell}[p,h] := \sum_{i=1}^g \frac{d^{\ell}}{dt^{\ell}} p(x_1,..., x_i + th,..., x_g).$$ A NC polynomial $p$ is said to be $\ell$-harmonic if $\lap_{\ell}[p,h] = 0$. When $\ell = 2$, then the $\ell$-Laplacian is simply called the Laplacian and a $\ell$-harmonic NC polynomial is simply called harmonic. More generally, the concept of a constant coefficien

Author

Nelson, Christopher

Published

2012

Language

EN