G{\aa}rding Polynomials

TL;DR

This paper introduces G{ {a}}rding polynomials, a new class of multivariate polynomials with invariant positivity regions, extending real stable polynomials and enabling new combinatorial and matrix applications.

Contribution

The paper defines G{ {a}}rding polynomials, proves their properties, and demonstrates their applications in matroid theory, graph theory, and matrix characteristic polynomials.

Findings

G{ {a}}rding polynomials extend real stable polynomials.

Multi-affine G{ {a}}rding polynomials with nonnegative coefficients satisfy the Rayleigh property.

Applications include new negative dependence results for matroids, graphs, and matrix characteristic polynomials.

Abstract

We introduce G{\aa}rding polynomials, a class of real multivariate polynomials characterized by positivity regions that are invariant under translation by positive vectors and closed under strictly positive affine transformations. We prove that this geometric formulation is equivalent both to a reduction to the multi-affine setting via polarization and to a recursive criterion in terms of partial derivatives. The class of G{\aa}rding polynomials strictly extends that of real stable polynomials while preserving many of their structural properties. In particular, multi-affine G{\aa}rding polynomials with nonnegative coefficients satisfy the Rayleigh property, and their positive univariate specializations have ultra log-concave coefficient sequences. The G{\aa}rding property for several matroid generating functions is preserved under natural matroid operations. As applications, we derive new negative dependence results for generating functions associated with various classes of matroids and graphs, including examples previously beyond the scope of real stability and Lorentzian methods. We further obtain analogous results for characteristic polynomials arising from certain matrix classes.