2024/12/04 by K D Johnson, Donald Richards, Johnson, Kenneth W. +1
Physics and Astronomy · #05E05 #15A72 #33C80 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Primary 33C20 #Quantum Mechanics and Applications #Secondary 15A15 #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.2412.03000
openalex publication_date 2024/12/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a given positive integer m, the concept of hyperdeterminantal total positivity is defined for a kernel K\colon \mathbb R2m → \mathbb R, thereby generalizing the classical concept of total positivity. Extending the fundamental example, K(x,y) = exp(xy), x, y ∈ ℝ, of a classical totally positive kernel, the hyperdeterminantal total positivity property of the kernel K(x1,…,x2m) = exp(x1⋯ x2m), x1,…,x2m ∈ ℝ is established. By applying Matsumoto's hyperdeterminantal Binet-Cauchy formula, we derive a generalization of Karlin's basic composition formula; then we use the generalized composition formula to construct several examples of hyperdeterminantal totally positive kernels. Further generalizations of hyperdeterminantal total positivity by means of the theory of finite reflection groups are described and some open problems are posed.