Criterions for the positive definiteness of real supersymmetric tensors

The positive definiteness of an even-degree homogeneous polynomial form f(x) plays an important role in the stability study of nonlinear autonomous systems via Lyapunov's direct method in automatic control, and the positive definiteness of f(x) is equivalent to that of an even-order supersymmetric tensor which defines f(x). In this paper, we provide some criterions for identifying the positive definiteness of an even-order real supersymmetric tensor. Moreover, an iterative algorithm for identifying the positive definiteness of an even-order real supersymmetric tensor is obtained. Numerical examples are given to verify the corresponding results.