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Oriented and Valuated Delta Matroids from Stable Polynomials
Abstract
Stable polynomials are the natural multivariate generalization of real rooted univariate polynomials. While their definition is purely algebraic in nature, they have deep connections to combinatorics. One such connection is the support theorem proved by Brändén in 2007, showing that the support of any stable polynomial is a jump system, and hence that the support of any multiaffine stable polynomial is a Δ-matroid. In this work, we generalize this result, showing that coefficients of multiaffine real stable polynomials give rise to oriented Δ-matroids, and coefficients of multiaffine stable polynomials over Puiseux series give rise to valuated Δ-matroids.
Related
- Valuated delta matroids and principal minors of Hermitian matrices
- A Fano framework for binary delta-matroids
- Presheaf structures governed by oriented matroids and pseudohyperplane arrangements
- Real-rootedness of the Z-polynomials of sparse paving matroids and strict interlacing for uniform matroids
- Real-rootedness of Kazhdan--Lusztig polynomials of sparse paving matroids
BibTeX
@misc{chin2026oriented,
title = {{Oriented and Valuated Delta Matroids from Stable Polynomials}},
author = {Tracy Chin},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13501},
eprint = {2609.13501},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13501},
}
RIS
TY - UNPB
TI - Oriented and Valuated Delta Matroids from Stable Polynomials
AU - Tracy Chin
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.13501
UR - https://arxiv.org/abs/2609.13501
AB - Stable polynomials are the natural multivariate generalization of real rooted univariate polynomials. While their definition is purely algebraic in nature, they have deep connections to combinatorics. One such connection is the support theorem proved by Brändén in 2007, showing that the support of any stable polynomial is a jump system, and hence that the support of any multiaffine stable polynomial is a Δ-matroid. In this work, we generalize this result, showing that coefficients of multiaffine real stable polynomials give rise to oriented Δ-matroids, and coefficients of multiaffine stable polynomials over Puiseux series give rise to valuated Δ-matroids.
ER -