【摘要】
Gårding polynomials provide a geometric framework for studying multivariate polynomials through their positivity regions. We develop a general theory of Gårding polynomials based on the positive ray test.
We introduce the class of ideal Gårding polynomials, a refinement that recursively incorporates convexity through derivatives. This yields strong structural consequences, including concavity properties and new inequalities. In the univariate setting, Gårding polynomials admit an integral representation determined by their root data, and their universal models arise as volume polynomials of certain polytopes. This establishes a new connection between positivity structures of polynomials and the convex geometry of polytopes, leading to new Newton–Maclaurin type inequalities.
A central application is a polarization theorem for ideal Gårding polynomials, which recovers and extends earlier results of Lin by a different method.
We also discuss applications in analysis and combinatorics. In analysis, we develop a geometric framework for Liouville rigidity for fully nonlinear elliptic equations, yielding a broad class of examples. In combinatorics, we show that many generating functions of matroids are Gårding, implying Rayleigh properties and ultra log-concavity, and suggesting new conjectures extending the strong Rayleigh program.
This work is joint with Biao Ma.
