Sharp Spectral Bounds for Symmetric Positive Definite Tensors via Multiple Algebraic Invariants

Abstract

We extend the trace--determinant framework of Nayak, Sharma, and Mishra~\cite{nayak2026} for bounding the H-eigenvalues of symmetric positive definite tensors. First, we replace the Arithmetic--Geometric Mean (AM--GM) relaxation underlying previous bounds by the exact solution of the associated constrained optimization problem, yielding sharp upper and lower bounds that are attained on the admissible spectral variety. Second, we incorporate higher-order power sums as additional spectral invariants and prove a structural theorem showing that any extremizer over a $K$-invariant feasibility region has at most $K$ distinct spectral values. This reduces the problem to a finite collection of low-dimensional polynomial systems and yields a hierarchy of increasingly tight bounds. For the four-invariant case $(T,S,p_3,D)$, we develop a complete theory including solution-count estimates, a multistart Newton algorithm, and sharpness conditions. We also derive closed-form bounds in small dimensions, establish perturbation estimates, and obtain refined Lyapunov region-of-attraction bounds. Numerical experiments for dimensions up to $d=100$ show that the sharp three-invariant bound reduces the median relative overestimation gap from $53\%$ to $6\%$ while maintaining low computational cost. The framework is validated on tensors with real H-spectrum.