A Real Counterexample to the Drury Permanent Conjecture

Abstract

The Drury permanent conjecture, published by Zhang in 2016 and disproved over the complex numbers by Hutchinson in 2017, has remained open in the real case, where a proof would imply the real case of the Chollet conjecture. The real 5 × 5 counterexample reported by Paksoy in 2023 concerns a different inequality, whose principal submatrices have order n −1 rather than order n−2 as in the conjecture; it is verified that Paksoy’s matrix satisfies the conjectured inequality strictly, under every simultaneous permutation of its rows and columns. The real case is then settled in the negative: an 8 × 8 real positive semidefinite matrix with integer entries and rank 3 is exhibited that violates the inequality, with an exact integer certificate of the violation. Direct sums with identity matrices give counterexamples of every order n ≥ 8.