Inductive Proof of Borchardt's Theorem

TL;DR

This paper presents an inductive proof of Borchardt's theorem, demonstrating a novel approach to calculating the permanent of Cauchy matrices and highlighting implications for antisymmetric geminal products in quantum chemistry.

Contribution

It introduces an inductive proof of Borchardt's theorem and explores its implications for the tractability of antisymmetric geminal product methods.

Findings

Inductive proof of Borchardt's theorem established

Cauchy matrix permanent calculation linked to auxiliary determinants

Implications for the computational tractability of APr2G methods

Abstract

We provide an inductive proof of Borchardt's theorem for calculating the permanent of a Cauchy matrix via the determinants of auxiliary matrices. This result has implications for antisymmetric products of interacting geminals (APIG), and suggests that the restriction of the APIG coefficients to Cauchy form (typically called APr2G) is special in its tractability.