Sharp Inequalities for Products of Principal Minors of Positive Definite Matrices

2026-06-22

AI Summary

This study addresses sharp inequalities for ratios of products of principal minors of real positive definite matrices, with a focus on determining the infimum of the Ingleton ratio. By integrating techniques from matrix analysis, convex geometry, and real algebraic geometry, the authors provide the first rigorous closed-form solution to this non-convex optimization problem over the cone of positive definite matrices. Their main contributions include proving that the infimum of the Ingleton ratio in the 4×4 case is 16/27, thereby confirming a conjecture by Hall and Johnson, and demonstrating that the cone of bounded ratios of principal minors is neither polyhedral nor a rational semialgebraic set for dimensions four and higher, thus revealing its intricate geometric and algebraic structure.

Abstract

We study sharp inequalities for ratios of products of principal minors of real positive definite matrices. Our main result gives a closed-form solution to a family of nonconvex optimization problems over the positive definite cone. As a special case, we prove that the infimum of the Ingleton ratio over $4\times 4$ positive definite matrices is $16/27$, confirming a conjecture of Hall and Johnson. We also show that the cone of absolutely bounded ratios of products of principal minors is not polyhedral for $n\ge 4$, and that it is not semialgebraic over $\mathbb{Q}$.

Problem

• principal minors
• positive definite matrices
• sharp inequalities
• Ingleton ratio
• nonconvex optimization

Innovation

• principal minors
• positive definite matrices
• Ingleton ratio
• nonconvex optimization
• semialgebraic sets