Characterizing Real-Valued Multivariate Complex Polynomials and Their Symmetric Tensor Representations

TL;DR

This paper explores conditions under which multivariate complex polynomials and their symmetric tensor representations are real-valued, introduces new eigenvalue concepts for complex tensors, and extends Banach's theorem to the complex setting.

Contribution

It provides new criteria for complex polynomials to be real-valued, introduces eigenvalue notions for complex tensors, and generalizes Banach's theorem to complex symmetric tensors.

Findings

Conditions for complex polynomials to be real-valued

New eigenvalue/eigenvector definitions for complex tensors

Extension of Banach's theorem to complex tensors

Abstract

In this paper we study multivariate polynomial functions in complex variables and the corresponding associated symmetric tensor representations. The focus is on finding conditions under which such complex polynomials/tensors always take real values. We introduce the notion of symmetric conjugate forms and general conjugate forms, and present characteristic conditions for such complex polynomials to be real-valued. As applications of our results, we discuss the relation between nonnegative polynomials and sums of squares in the context of complex polynomials. Moreover, new notions of eigenvalues/eigenvectors for complex tensors are introduced, extending properties from the Hermitian matrices. Finally, we discuss an important property for symmetric tensors, which states that the largest absolute value of eigenvalue of a symmetric real tensor is equal to its largest singular value; the result is known as Banach's theorem. We show that a similar result holds in the complex case as well.