A division theorem for real analytic functions

Abstract

AbstractWe characterize those homogeneous polynomials P e ℂ[z 1 , … , z d ] for which the principal ideal (P) = P · A(ℝ d ) is complemented in A(ℝ d ) or, equivalently, those which admit a continuous linear division operator. The condition is the same as that which characterizes, among the homogeneous polynomials, those which are nonelliptic and for which P(D) is surjective in A(ℝ d ), and those for which P(D) admits a continuous linear right inverse in C ∞ (ℝ d ). It depends only on the type of real singularities.