Ultrastability of ideals of homogeneous polynomials and multilinear mappings on Banach spaces

Using the theory of full and symmetric tensor norms on normed spaces, a theorem of Kursten and Heinrich on ultrastability and maximality of normed operator ideals is extended to ideals of n-homogeneous polynomials and n-linear mappings-scalar-valued and vector-valued. The motivation for these results is the following important special case: the "uniterated" Aron-Berner extension (q) over bar (U) : E" --> F" of an n-homogeneous polynomial q : E-->F to the bidual remains in certain ideals under preservation of the norm. Moreover, Lotz's characterization of maximal normed ideals of linear mappings through appropriate tensor norms is proved for ideals of n-homogeneous scalar-valued polynomials and ideals of n-linear mappings.