  The paper introduces a notion of the Laplace operator of a polynomial p in
noncommutative variables x=(x_1,...,x_g). The Laplacian Lap[p,h] of p is a
polynomial in x and in a noncommuting variable h. When all variables commute we
have Lap[p,h]=h^2\Delta_x p where \Delta_x p is the usual Laplacian. A
symmetric polynomial in symmetric variables will be called harmonic if
Lap[p,h]=0 and subharmonic if the polynomial q(x,h):=Lap[p,h] takes positive
semidefinite matrix values whenever matrices X_1,..., X_g, H are substituted
for the variables x_1,...,x_g, h. In this paper we classify all homogeneous
symmetric harmonic and subharmonic polynomials in two symmetric variables. We
find there are not many of them: for example, the span of all such subharmonics
of any degree higher than 4 has dimension 2 (if odd degree) and 3 (if even
degree). Hopefully, the approach here will suggest ways of defining and
analyzing other partial differential equations and inequalities.
