Timeline for Stability of real polynomials with positive coefficients
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9 events
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| Dec 30, 2022 at 15:28 | comment | added | Colin Tan | The writeup that Theorem A is a corollary of Catlin and D'Angelo's result is given here: arxiv.org/abs/1701.02040 | |
| Apr 13, 2017 at 12:57 | history | edited | CommunityBot |
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| Sep 19, 2014 at 4:38 | comment | added | Alexandre Eremenko | @Colin Tan: we knew about Catlin and d'Angelo results, and even discussed "our" theorem A with each of them. We could not derive Theorem A from their statements. I mean the relation of the condition about "first two and last two" with their conditions. | |
| Sep 18, 2014 at 13:56 | comment | added | user2529 | In particular, as you noted, the condition that $|f(z)|<f(|z|)$ is equivalent to the associated Hermitian metric of $f$ satisfying the Sharp Global Cauchy Schwar inequality. The other condition that the first two and the last two coefficients of $f$ are strictly positive is equivalent to its associated globalizable Hermitian metric being negatively curved. | |
| Sep 18, 2014 at 13:37 | comment | added | user2529 | can be isometrically embedded into complex projective space. Theorem A can be deduced from this result of Catlin and D'Angelo in the case where the complex compact manifold is the Riemann sphere and using the fact that every negative holomorphic line bundle over the Riemann surface is antiample. | |
| Sep 18, 2014 at 13:34 | comment | added | user2529 | After reading the answers by you and Prof Handleman and having a discussion with my PhD supervisor, we realized that the affirmative answer to my question and in fact what is now Theorem A in your revised preprint is a corollary of an isometric embedding theorem of Catlin and D'Angelo, MRL 1999. There, Catlin and D'Angelo proved, in particular, that for a globalizable Hermitian metric on a holomorphic line bundle over a compact complex manifold, if this metric is negatively curved and satisfies the so called Sharp Global Cauchy Schwarz inequality, then sufficiently large powers of this metric | |
| Sep 16, 2014 at 13:51 | history | edited | Alexandre Eremenko | CC BY-SA 3.0 |
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| Sep 10, 2014 at 17:48 | history | edited | Alexandre Eremenko | CC BY-SA 3.0 |
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| Sep 10, 2014 at 2:40 | history | answered | Alexandre Eremenko | CC BY-SA 3.0 |