稳定多项式上的定向与赋值的Delta拟阵
Oriented and Valuated Delta Matroids from Stable Polynomials
AI总结：
本文推广Brändén的支撑定理，证明多仿射实稳定多项式的系数产生定向$\Delta$-拟阵，Puiseux级数上的多仿射稳定多项式的系数产生赋值的$\Delta$-拟阵。
AI中文摘要：
稳定多项式是实根单变量多项式的自然多元推广。虽然其定义本质上是代数的，但它们与组合学有深刻的联系。其中一个联系是Brändén在2007年证明的支撑定理，表明任何稳定多项式的支撑是一个跳跃系统，因此任何多仿射稳定多项式的支撑是一个$\Delta$-拟阵。在这项工作中，我们推广了这一结果，表明多仿射实稳定多项式的系数产生定向$\Delta$-拟阵，而Puiseux级数上的多仿射稳定多项式的系数产生赋值的$\Delta$-拟阵。
英文摘要：
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.
