  We give a fast, exact algorithm for solving Dirichlet problems with
polynomial boundary functions on quadratic surfaces in R^n such as ellipsoids,
elliptic cylinders, and paraboloids. To produce this algorithm, first we show
that every polynomial in R^n can be uniquely written as the sum of a harmonic
function and a polynomial multiple of a quadratic function, thus extending a
theorem of Ernst Fischer. We then use this decomposition to reduce the
Dirichlet problem to a manageable system of linear equations. The algorithm
requires differentiation of the boundary function, but no integration. We also
show that the polynomial solution produced by our algorithm is the unique
polynomial solution, even on unbounded domains such as elliptic cylinders and
paraboloids.
