Chollet’s conjecture

← Minc’s list

Minc’s list · Conjecture 30

Chollet’s conjecture

Open

Posed by
Chollet
First posed
1982

Statement

Posed by J. Chollet in 1982, in the American Mathematical Monthly, as a permanental analogue of Oppenheim’s inequality; it is Conjecture 30 in Minc’s 1983 survey.

Conjecture 30. If \(A\) and \(B\) are \(n\times n\) positive semidefinite Hermitian matrices, then

\[ \operatorname{per}(A\circ B)\ \le\ \operatorname{per}A\,\operatorname{per}B, \]

where \(A\circ B=(a_{ij}b_{ij})\) is the Hadamard (entrywise) product.

Chollet observed that the conjecture is equivalent to its special case \(\operatorname{per}(A\circ\overline A)\le(\operatorname{per}A)^2\), where \(\overline A\) is the entrywise conjugate.

Prior progress

  • The cases \(n=2,3\) were proved by Gregorac and Hentzel (1987); the case \(n=3\) also follows from Bapat and Sunder’s work on Soules’s conjecture.
  • The case \(n=4\) was proved by Hutchinson (2021) for real matrices and by Rodtes (2024) for complex ones.
  • Further special classes were settled by Sa-nguansin and Rodtes (2025), and by Pant and Singh (2026) for classes of matrices with bipartite support, including graph Laplacians.
  • After we started, Li, Zhao and Luo posted a proof of the full inequality for every \(n\le6\) (preprint, 31 August 2026). We reconstructed their proof independently and found no gap.

Our progress so far

  • Rank at most two, every order. If \(A\) and \(B\) both have rank at most \(2\), then \(\operatorname{per}(A\circ B)\le\operatorname{per}A\operatorname{per}B\) for every \(n\).
  • Rank at most three. The inequality holds when both matrices have rank at most \(3\), for every \(n\ne7\) in the complex case and for every \(n\) in the real case. Orders up to \(6\) follow from the small-order results above.
  • Alternative proofs at orders four and five. Our paper A Conjecture on the Permanent of Positive Semidefinite Matrices states a quadratic inequality that implies Chollet’s inequality at each order, and proves it through order five. This gives an alternative proof of the order-five case and, through the order-four case of Soules’ permanent-on-top conjecture settled there, another proof of the complex order-four case.

References

  1. J. Chollet, Is there a permanental analogue to Oppenheim’s inequality?, Amer. Math. Monthly 89 (1982), 57–58. doi:10.1080/00029890.1982.11995380
  2. R. J. Gregorac and I. R. Hentzel, A note on the analogue of Oppenheim’s inequality for permanents, Linear Algebra Appl. 94 (1987), 109–112.
  3. R. B. Bapat and V. S. Sunder, An extremal property of the permanent and the determinant, Linear Algebra Appl. 76 (1986), 153–163. doi:10.1016/0024-3795(86)90220-X
  4. G. Hutchinson, An elementary proof of Chollet’s permanent conjecture for 4 × 4 real matrices, Spec. Matrices 9 (2021), 83–102. doi:10.1515/spma-2020-0126
  5. K. Rodtes, Chollet’s permanent conjecture for 4 × 4 matrices, Linear and Multilinear Algebra 72 (2024), 2633–2638. doi:10.1080/03081087.2023.2279150
  6. S. Sa-nguansin and K. Rodtes, Permanents of correlation matrices and the Chollet permanental conjecture, Linear and Multilinear Algebra 73 (2025), 4084–4096. doi:10.1080/03081087.2025.2588584
  7. P. Pant and R. Singh, Structural classes for Chollet’s permanent conjecture, preprint (2026). arXiv:2604.24192
  8. Q. Li, Q. Zhao and W. Luo, Chollet’s permanent conjecture through order six via border induction, preprint (2026). Preprints.org 202608.2196
  9. F. Zhang, An update on a few permanent conjectures, Spec. Matrices 4 (2016), 305–316. doi:10.1515/spma-2016-0030