Marcus–Minc transform inequality

← Minc’s list

Minc’s list · Conjecture 3

Marcus–Minc transform inequality

Proved by us arXiv preprint, September 2026

Posed by
Marcus and Minc
First posed
1967

Statement

Posed by M. Marcus and H. Minc in 1967, in a paper on van der Waerden’s conjecture; it is Conjecture 3 in Minc’s catalogue.

Conjecture 3. Let \(\Omega_n\) be the set of \(n\times n\) doubly stochastic matrices and let \(J_n\) be the \(n\times n\) matrix with every entry \(1/n\). If \(A\in\Omega_n\) and \(n\ge 2\), then

\[ \operatorname{per}A\;\ge\;\operatorname{per}\!\left(\frac{nJ_n-A}{n-1}\right). \]

If \(n\ge 4\), equality holds if and only if \(A=J_n\).

The matrix \((nJ_n-A)/(n-1)\) has entries \((1-a_{ij})/(n-1)\), so it is again doubly stochastic. Marcus and Minc originally claimed uniqueness of equality already for \(n\ge3\); Wang later found further equality cases in order three, which is why the catalogue states the equality clause for \(n\ge4\).

Resolution

We proved the conjecture in every order \(n\ge2\) and determined all cases of equality:

  • \(n=2\): every \(A\in\Omega_2\);
  • \(n=3\): \(J_3\) and the six matrices \((\mathbf 1\mathbf 1^{\mathsf T}-P)/2\), where \(P\) is a permutation matrix;
  • \(n\ge4\): only \(A=J_n\).

The proof combines two global estimates in terms of \(r=\lVert A-J_n\rVert_F\). A lower bound

\[ \operatorname{per}A\ \ge\ \frac{n!}{n^n}\exp\!\left(\Bigl(\frac1{2n}+\frac1{3n^2}\Bigr)r^2\right) \]

comes from Gurvits’s capacity inequality for real stable polynomials, and a matching upper bound for \(\operatorname{per}\bigl((nJ_n-A)/(n-1)\bigr)\) comes from a sharp estimate for sums of subpermanents of matrices with zero line sums. For \(n\ge4\) this even gives a quantitative gap, \(\operatorname{per}A-\operatorname{per}\bigl((nJ_n-A)/(n-1)\bigr)\ge \frac{n!}{n^n}\,\epsilon_n r^2\) with an explicit \(\epsilon_n>0\).

Read our paper → arXiv:2609.29262 ↗

Earlier progress

  • Marcus and Minc (1967) proved the inequality for \(n=2\), for positive semidefinite symmetric \(A\), and for \(A\) sufficiently close to \(J_n\).
  • Wang (1977) proved the case \(n=3\) and found the additional equality cases; Foregger (1979) proved the case \(n=4\).
  • Chang (1983) proved it for all \(A\) outside a sufficiently large neighbourhood of \(J_n\).
  • Hwang (1989) proved it for partly decomposable \(A\), and Malek (1989) for normal \(A\) whose eigenvalues lie in the sector \(\lvert\arg z\rvert\le\pi/(2n)\).

References

  1. M. Marcus and H. Minc, On a conjecture of B. L. van der Waerden, Proc. Cambridge Philos. Soc. 63 (1967), 305–309. doi:10.1017/S0305004100041219
  2. E. T. H. Wang, On a conjecture of M. Marcus and H. Minc, Linear and Multilinear Algebra 5 (1977), 145–148. doi:10.1080/03081087708817189
  3. T. H. Foregger, Remarks on a conjecture of M. Marcus and H. Minc, Linear and Multilinear Algebra 7 (1979), 123–126. doi:10.1080/03081087908817268
  4. D. K. Chang, A note on a conjecture of M. Marcus and H. Minc, Linear and Multilinear Algebra 13 (1983), 115–117. doi:10.1080/03081088308817511
  5. S.-G. Hwang, Some permanental inequalities, Bull. Korean Math. Soc. 26 (1989), 35–42.
  6. M. Malek, A note on a permanental conjecture of M. Marcus and H. Minc, Linear and Multilinear Algebra 25 (1989), 71–73. doi:10.1080/03081088908817929
  7. L. Gurvits, Van der Waerden/Schrijver–Valiant like conjectures and stable (aka hyperbolic) homogeneous polynomials: one theorem for all, Electron. J. Combin. 15 (2008), R66. arXiv:0711.3496
  8. Y. Lavi, The Marcus–Minc transform inequality, preprint (2026). arXiv:2609.29262