Sum Of Even Powers Of Real Linear Forms

This work initiates a systematic analysis of the representation of real forms of even degree as sums of powers of linear forms and the resulting implications in real algebraic geometry, number theory, combinatorics, functional analysis, and numerical analysis. This subject is closely related to the study of positive semidefinite forms and their representation of sums of squares of forms, begun by Hilbert and revived by Motzkin, R M Robinson, Choi, and Lam. Under suitable reinterpretation, some of the results can be traced to work of Akhiezer and Krein in the classical moment problem. Since the time of Liouville, representations of the form hn,2s=(x+21+.+x2n)s as a sum of 2s-th powers have been used in the study of Waring's problem, and they are equivalent to certain Banach space embeddings. The proofs utilize elementary techniques from linear algebra, convexity, number theory, and real algebraic geometry and many explicit examples and relevant historical remarks are presented.