Linear Preservers of Stable and Lorentzian Polynomials and Deformations of Hyperbolicity Cones
summary
Linear operators on bounded-degree polynomials preserve complex stability exactly when their algebraic symbol criterion is stable, and that extends to infinite-dimensional operators via the Laguerre–Pólya transcendental symbol; homogeneous stable symbols are automatically Lorentzian so operators with Lorentzian ↔ polymatroid + D−2 derivatives stable symbols preserve the Lorentzian property. Key constructions include Renegar derivatives, box truncations, and convolutions that track the max root or hyperbolicity set; the symmetric additive convolution unifies Marcus–Spielman–Srivastava-type bounds and any linear operator can be realized as such a convolution.
AI-written summary · about
chapters (12)
- 0:00Start
- 1:41Prelude
- 2:46Stable polynomials
- 5:31Brief history
- 9:11Stability preservers
- 12:01Pólya-Schur master theorem
- 14:04Transcendental characterization
- 20:34Discrete convexity
- 21:56Lorentzian (CLC, SLC) polynomials
- 24:08Lorentzian polynomials
- 27:47Lorentzian preservers
- 31:42How do zeros move under preservers?
related talks
53mConnection Laplacian, Hodge Laplacian, and Tensor Laplacian of a Graph
54mNegative Dependence and Lorentzian Polynomials
43mApplied Hodge Theory
01h 11m(Hidden) Symmetries of Computational Problems (continued)
01h 00mLog-concave Polynomials and Spectral Independence 2
01h 00mAverage-case Complexity for Polynomials, and All That
01h 11mPedagogical Talk: Failure of Low-Degree Polynomials
01h 09mOptimization and Sampling Under Symmetry: Part 2 -- Lie Theory