Completely Log-Concave Polynomials

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" summary

Defines completely log-concave polynomials as those positive on a convex cone whose every directional derivative (along cone directions) is log‑concave, and shows stability on the nonnegative orthant implies this property. Key consequences: closure under directional derivatives and initial forms; Newton polytope must be a generalized permutahedron with support filling all integer points; all matroid basis-generating polynomials (including the Fano) are included though not all are real stable. Gives a finite test by reducing to checking certain quadratics have one positive eigenvalue.

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" chapters (13)

1. 0:00Definitions and motivation
2. 2:59Completely log-concave
3. 4:54Examples and properties
4. 8:21Relation to stability
5. 10:21Symmetric polynomials
6. 13:08Initial forms
7. 16:32Newton polytopes
8. 22:04Matroids and polynomials
9. 26:04Generalized permutahedra
10. 28:43Applications to algorithms
11. 33:48Testing concavity
12. 39:33Necessary conditions
13. 45:20Future directions

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