Timeline for Differential operators that preserve real-rootedness

7 events

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Apr 13, 2017 at 12:58	history	edited	CommunityBot		replaced http://mathoverflow.net/ with https://mathoverflow.net/
Jul 17, 2012 at 17:49	comment	added	David E Speyer		Thanks! The authors state (last paragraph, section 4) that an answer to the question about closed intervals would "have many interesting applications". Sounds like this is open for the time being.
Jul 17, 2012 at 17:22	comment	added	Gjergji Zaimi		Notice that the corresponding classification problem for positive reals is open.
Jul 17, 2012 at 17:07	comment	added	Gjergji Zaimi		Also look at their Annals paper annals.math.princeton.edu/2009/170-1/p14 Their main result characterizes linear operators which preserve real rooted polynomials as the ones whose symbol is a real stable polynomial. The latter can be further characterized as a determinant by another theorem of the authors.
Jul 17, 2012 at 15:53	comment	added	David E Speyer		Nope, didn't know that. Thanks! I wouldn't be surprised if there were some clever trick to turn one into the other.
Jul 17, 2012 at 15:50	comment	added	Dan Petersen		Perhaps you know this, but the if you had demanded instead that $\mathcal D$ preserves the property of having all roots in $\mathbb R$, then the result would be one of the main theorems in Borcea and Brändén, Multivariate Pólya-Schur classification problems in the Weyl algebra, Proc. Lond. Math. Soc. (3) 101 (2010).
Jul 17, 2012 at 15:14	history	asked	David E Speyer	CC BY-SA 3.0