For a complex polynomial \(P\) in \(d\) variables, the principal ideal \( (P) \) in the space \( \mathcal A(\mathbb{R}^d) \) of all real analytic functions on \( \mathbb{R}^d \) is studied. The question is, whether there exists a continuous linear operator such that \( T(Pf) = f \). One main result states that for the existence of such an operator it is necessary that the variety \(V\) of \(P\) satisfies the local Phragmén-Lindelöf condition at each of its real points. This condition came up first in L. Hörmander’s characterization of surjective partial differential operators on \( \mathcal A(\mathbb{R}^d) \) [Invent. Math. 21, 151–182 (1973; Zbl 0282.35015)]. The converse of this result is shown if either \( V \) is homogeneous or \( V \cap \mathbb{R}^d \) is compact.
Necessity of the Phragmén-Lindelöf condition is proved using the theory of (PLB)-spaces, developed by P. Domański and D. Vogt [Stud. Math. 140, 57–77 (2000; Zbl 0973.46067)]. In the compact case, sufficiency follows directly from Vogt’s characterization of extension operators for real analytic functions on real analytic subvarieties of \( \mathbb{R}^d \) [J. Reine Angew. Math. 606, 217–233 (2007; Zbl 1133.46014)]. In the homogeneous case, an extension operator near zero and one near infinity are patched together using Cartan-Oka-theory.