Positively Hyperbolic Varieties
summary
Defines positively hyperbolic varieties via imaginary-part sign conditions and proves algebraic matroids of such varieties are positroids, using sign-variation theorems and tropicalization. Main claim: any maximal cell of trop(X) is spanned by 0/1/−1 vectors whose supports are disjoint and non-crossing supports, so initial degenerations reduce to toric cases. Corollaries: reciprocal linear spaces and many curves give examples, Bergman fans of positroids occur, and Chow polytopes become generalized permutohedra.
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chapters (11)
- 0:00Introduction
- 1:51Hyperbolic and stable polynomials
- 3:39Hyperbolic varieties
- 6:49Positively hyperbolic varieties
- 8:58Matroid structures
- 13:09Tropicalization
- 25:44Algebraic matroids and positroids
- 33:21Tropicalization structure theorem
- 41:33Realizability questions
- 45:05Open problems
- 48:59Q&A
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