Unique Minimizers for Permanents, Mixed Discriminants, and Log-concave Polynomials
Abstract
The permanent and mixed discriminant of positive matrices are classic problems for which we do not expect an efficient algorithm for exact computation. Thus much work has been done to understand how well we can bound and approximately compute these quantities. One line of research in this area begins with the results of the first author, where van der Waerden lower bounds of n!/nn are proven for doubly stochastic inputs for both problems, using a simple proof via stable polynomials. Along with the bound itself, the same techniques are used to show that the permanent and mixed discriminant are uniquely minimized at a certain natural symmetric input. In this paper, we generalize those results in two ways. First, we extend the unique minimization results beyond doubly stochastic inputs to other marginals which are near doubly stochastic. This yields the first such unique minimization results for the mixed discriminant beyond the doubly stochastic case. We also discuss why one cannot hope similar results to hold in general for all marginals. Second, we extend the unique minimization result for real stable polynomials to strongly log-concave (aka Lorentzian) polynomials in the doubly stochastic case. This captures an analogous previous result on unique minimization for the mixed volume. Finally, we discuss various open problems related to these results.
Related
- Chow polynomials and chain polynomials of matroids are not log-concave
- Méthodes à ε-près pour des systèmes modélisant des gaz permanents
- A Koteljanskii inequality for permanents
- Discriminants of Hermitian forms, maximal degenerations and Kontsevich's tropical approach to the Hodge conjecture
- Inequalities for rank-two permanents and finite free convolutions
BibTeX
@misc{gurvits2026unique,
title = {{Unique Minimizers for Permanents, Mixed Discriminants, and Log-concave Polynomials}},
author = {Leonid Gurvits and Jonathan Leake},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.21083},
eprint = {2609.21083},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21083},
}
RIS
TY - UNPB
TI - Unique Minimizers for Permanents, Mixed Discriminants, and Log-concave Polynomials
AU - Leonid Gurvits
AU - Jonathan Leake
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2609.21083
UR - https://arxiv.org/abs/2609.21083
AB - The permanent and mixed discriminant of positive matrices are classic problems for which we do not expect an efficient algorithm for exact computation. Thus much work has been done to understand how well we can bound and approximately compute these quantities. One line of research in this area begins with the results of the first author, where van der Waerden lower bounds of n!/n^n are proven for doubly stochastic inputs for both problems, using a simple proof via stable polynomials. Along with the bound itself, the same techniques are used to show that the permanent and mixed discriminant are uniquely minimized at a certain natural symmetric input. In this paper, we generalize those results in two ways. First, we extend the unique minimization results beyond doubly stochastic inputs to other marginals which are near doubly stochastic. This yields the first such unique minimization results for the mixed discriminant beyond the doubly stochastic case. We also discuss why one cannot hope similar results to hold in general for all marginals. Second, we extend the unique minimization result for real stable polynomials to strongly log-concave (aka Lorentzian) polynomials in the doubly stochastic case. This captures an analogous previous result on unique minimization for the mixed volume. Finally, we discuss various open problems related to these results.
ER -