HEXAGONbeta

A Dittert-like inequality for homogeneous stable polynomials

A mix of human-written and AI-generated textHuman understanding: all partsmath.CO — Combinatoricsmath.RA — Rings and Algebras

Contributed by Yair Lavi ↗

Version 1 / Nov 10, 2026 / CC BY 4.0

Abstract

We prove a coefficient inequality for every homogeneous real stable polynomial P(s,x1,…,xn)P(s,x_1,\ldots,x_n) of degree n≥1n\ge1 with nonnegative coefficients. If P(1,0,…,0)=1P(1,0,\ldots,0)=1, ri=∂xiP(1,0,…,0)r_i=\partial_{x_i}P(1,0,\ldots,0), and ∑iri=n\sum_i r_i=n, the following inequality holds, where brackets denote coefficient extraction: [x1⋯xn]P(0,x1,…,xn)+2−∏iri−P(0,1,…,1)≥n!nn[x_1\cdots x_n]P(0,x_1,\ldots,x_n)+2-\prod_i r_i-P(0,1,\ldots,1)\ge \frac{n!}{n^n}. Equality holds exactly for P=(s+∑ixi/n)nP=(s+\sum_i x_i/n)^n. The normalizations can be removed by rescaling PP and the xix_i. We give applications to permanents, mixed discriminants, subpermanents, and stable combinatorial measures. In particular, the theorem gives an independent proof of Dittert's conjecture for nonnegative matrices in every dimension.

Tools used

OpenAI
GPT-5.6-sol
OpenAI
GPT-6 Astra
Anthropic
Claude FableVersion 5
Anthropic
Claude FableVersion 5.1

References

  1. R. B. Bapat, Mixed discriminants of positive semidefinite matrices , Linear Algebra and its Applications 126 (1989), 107--124. doi:10.1016/0024-3795(89)90009-8 (https://doi.org/10.1016/0024-3795(89)90009-8).DOI
  2. J. Borcea, P. Brändén, and T. M. Liggett, Negative dependence and the geometry of polynomials , Journal of the American Mathematical Society 22 (2009), 521--567. doi:10.1090/S0894-0347-08-00618-8 (https://doi.org/10.1090/S0894-0347-08-00618-8); arXiv:0707.2340v2 (https://arxiv.org/abs/0707.2340v2).arXivDOI
  3. G.-S. Cheon and I. M. Wanless, Some results towards the Dittert conjecture on permanents , Linear Algebra and its Applications 436 (2012), 791--801. doi:10.1016/j.laa.2010.08.041 (https://doi.org/10.1016/j.laa.2010.08.041).DOI
  4. Y.-B. Choe, J. G. Oxley, A. D. Sokal, and D. G. Wagner, Homogeneous multivariate polynomials with the half-plane property , Advances in Applied Mathematics 32 (2004), 88--187. doi:10.1016/S0196-8858(03)00078-2 (https://doi.org/10.1016/S0196-8858(03)00078-2); arXiv:math/0202034v2 (https://arxiv.org/abs/math/0202034v2).arXivDOI
  5. X. Dai, W. Deng, Y. Shi, T. Wu, and Y. Yang, A proof of the Dittert conjecture via boundary rigidity (2026). arXiv:2610.10545v1 (https://arxiv.org/abs/2610.10545v1).arXiv
  6. G. P. Egorychev, The solution of van der Waerden's problem for permanents , Advances in Mathematics 42 (1981), 299--305. doi:10.1016/0001-8708(81)90044-X (https://doi.org/10.1016/0001-8708(81)90044-X).DOI
  7. D. I. Falikman, Proof of the van der Waerden conjecture regarding the permanent of a doubly stochastic matrix , Mathematical Notes 29 (1981), 475--479. doi:10.1007/BF01163285 (https://doi.org/10.1007/BF01163285).DOI
  8. S. Friedland, A proof of a generalized van der Waerden conjecture on permanents , Linear and Multilinear Algebra 11 (1982), 107--120. doi:10.1080/03081088208817436 (https://doi.org/10.1080/03081088208817436).DOI
  9. L. Gårding, An inequality for hyperbolic polynomials , Journal of Mathematics and Mechanics 8 (1959), 957--965. doi:10.1512/IUMJ.1959.8.58061 (https://doi.org/10.1512/IUMJ.1959.8.58061).DOI
  10. L. Gurvits, Combinatorics hidden in hyperbolic polynomials and related topics (2004). arXiv:math/0402088 (https://arxiv.org/abs/math/0402088).arXiv
Show all 23 referencesShow fewer references
  1. L. Gurvits, The van der Waerden conjecture for mixed discriminants , Advances in Mathematics 200 (2006), 435--454. doi:10.1016/j.aim.2004.12.002 (https://doi.org/10.1016/j.aim.2004.12.002).DOI
  2. L. Gurvits, Van der Waerden/Schrijver--Valiant like conjectures and stable (aka hyperbolic) homogeneous polynomials: one theorem for all , Electronic Journal of Combinatorics 15 (2008), R66. doi:10.37236/790 (https://doi.org/10.37236/790); arXiv:0711.3496v2 (https://arxiv.org/abs/0711.3496v2).arXivDOI
  3. J. W. Helton and V. Vinnikov, Linear matrix inequality representation of sets , Communications on Pure and Applied Mathematics 60 (2007), 654--674. doi:10.1002/cpa.20155 (https://doi.org/10.1002/cpa.20155).DOI
  4. S.-G. Hwang, A note on a conjecture on permanents , Linear Algebra and its Applications 76 (1986), 31--44. doi:10.1016/0024-3795(86)90212-0 (https://doi.org/10.1016/0024-3795(86)90212-0).DOI
  5. S.-G. Hwang, On a conjecture of E. Dittert , Linear Algebra and its Applications 95 (1987), 161--169. doi:10.1016/0024-3795(87)90032-2 (https://doi.org/10.1016/0024-3795(87)90032-2).DOI
  6. B. Kafidov, Dittert's conjecture in dimension 16 via a joint-deficit scaling lemma (2026). arXiv:2607.19439 (https://arxiv.org/abs/2607.19439).arXiv
  7. A. S. Lewis, P. A. Parrilo, and M. V. Ramana, The Lax conjecture is true , Proceedings of the American Mathematical Society 133 (2005), 2495--2499. doi:10.1090/S0002-9939-05-07752-X (https://doi.org/10.1090/S0002-9939-05-07752-X).DOI
  8. H. Minc, Theory of permanents 1978--1981 , Linear and Multilinear Algebra 12 (1983), 227--263. doi:10.1080/03081088308817488 (https://doi.org/10.1080/03081088308817488).DOI
  9. P. P. M. do Nascimento, Complete working-proof chain and exact verification materials for Dittert's conjecture (assembled 25 July 2026). github.com/pedromnasc/dittert-conjecture-proof (https://github.com/pedromnasc/dittert-conjecture-proof/tree/894066bbaa715138c98bf3cb7c6fdb4f39a37701), commit 894066bbaa71 (accessed 29 October 2026).link
  10. Z. Pang, Proof of Dittert's conjecture for dimensions n≥17n\ge17 (2026). arXiv:2606.01531 (https://arxiv.org/abs/2606.01531).arXiv
  11. R. Sinkhorn, A problem related to the van der Waerden permanent theorem , Linear and Multilinear Algebra 16 (1984), 167--173. doi:10.1080/03081088408817620 (https://doi.org/10.1080/03081088408817620).DOI
  12. H. Tverberg, On the permanent of a bistochastic matrix , Mathematica Scandinavica 12 (1963), 25--35. doi:10.7146/math.scand.a-10669 (https://doi.org/10.7146/math.scand.a-10669).DOI
  13. D. G. Wagner and Y. Wei, A criterion for the half-plane property , Discrete Mathematics 309 (2009), 1385--1390. doi:10.1016/j.disc.2008.02.005 (https://doi.org/10.1016/j.disc.2008.02.005); arXiv:0709.1269v1 (https://arxiv.org/abs/0709.1269v1); corrigendum, Discrete Mathematics 313 (2013), 830 (https://doi.org/10.1016/j.disc.2012.11.001).arXivDOI

Version history

  1. v1 · 25 pagesSubmitted by Yair LaviInitial depositCurrentNov 07, 2026