An algebraic approach to solving boundary value problems

Let P(y(1),..., y(n)), Q(y(1),..., y(n)) be polynomials in R[y(1),..., y(n)] and let Q = Z(Q(y(1),..., y(n))) = {(r(1),..., r(n)) is an element of R(n) |Q(r(1),..., r(n) ) = 0} be the real algebraic set associated with Q(y(1),..., y(n)) and let (Q) over cap be a compact subset of the algebraic set Q. We describe an algebraic approach for solving the general boundary value problem (BVP): given partial differential equation (PDE) P((partial derivative/partial derivative x(1),..., (partial derivative/partial derivative x(n))) and a continuous function q : (Q) over cap -> R, find u(x(1),..., x(n)) is an element of R[[x(1),..., x(n)]] so that P(partial derivative/partial derivative x(1),..., partial derivative/partial derivative x(n))u(x(1),..., x(n)) = 0 and u(x(1),..., x(n))|(Q) over cap = q. We will show how the general technique applies in the case that P(y) is a homogeneous polynomial of degree deg(P(y)) and Q(y) =P(y) + (P) over cap (y), where (P) over cap (y) is a polynomial having deg((P) over cap (y)) < deg(P(y)) and prove that the solution is unique in this case. This article brings together ideas from partial differential equations, a generalization of the theory of functions of a complex variable and the theory of commutative algebras.