Characterizing real-valued multivariate complex polynomials and their symmetric tensor representations

In this paper we study multivariate polynomial functions in complex variables and their corresponding symmetric tensor representations. The focus is to find conditions under which such complex polynomials always take real values. We introduce the notion of symmetric conjugate forms and general conjugate forms, characterize the conditions for such complex polynomials to be real valued, and present their corresponding tensor representations. New notions of eigenvalues/eigenvectors for complex tensors are introduced, extending similar properties from the Hermitian matrices. Moreover, we study a property of the symmetric tensors, namely, the largest eigenvalue (in the absolute value sense) of a real symmetric tensor is equal to its largest singular value; the result is also known as Banach’s theorem. We show that a similar result holds for the complex case as well. Finally, we discuss some applications of the new notion of eigenvalues/eigenvectors for the complex tensors.