A Power Geometry and Hypergeometric Functions

Abstract

This paper is a corrected English translation of an original Japanese work introducing "power geometry," a research programme centred on power transformations, deformations and dualities of functions in the sense of optimisation theory. Its two main motivations are the origin of probability distributions (via maximum-likelihood principles, rational-function parameters and ODEs linked to information geometry, the Pearson system and Schwarzian equations) and the construction of matrix dynamical systems that compute eigenvalues, singular values and principal or minor subspaces. Building on Brockett's double-bracket flows, the author extends these systems to rectangular matrices, flag manifolds and difference-of-convex potentials, and shows how a continuous power parameter can interpolate between largest- and smallest-eigenvalue problems. Dual potentials, topological distinctions between principal and minor subspace flows, and links via matrix Schwarzian--Riccati equations are examined, with a view toward design principles for classical and quantum information-processing algorithms.