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We analyze metrics for how close an entire function of genus one is to having only real roots. These metrics arise from truncated Hankel matrix positivity-type conditions built from power series coefficients at each real point. Specifically, if such a function satisfies our positivity conditions and has well-spaced zeros, we show that all of its zeros have to (in some explicitly quantified sense) be far away from the real axis. The obvious interesting example arises from the Riemann zeta function, where our positivity conditions yield a family of relaxations of the Riemann hypothesis. One might guess that as we tighten our relaxation, the zeros of the zeta function must be close to the critical line. We show that the opposite occurs: any poten- tial non-real zeros are forced to be farther and farther away from the critical line.
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On two notions of total positivity for partial flag varieties
Lusztig’s total positivity implies Plücker positivity, and it is natural to ask when these two notions of positivity agree. Rietsch (2009) proved that they agree in the case of the Grassmannian Flk;n, and Chevalier (2011) showed that the two notions are distinct for Fl1,3;4. We show that in general, the two notions agree if and only if k1, ..., kl are consecutive integers.
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- Award ID(s):
- 2103026
- PAR ID:
- 10415662
- Date Published:
- Journal Name:
- Advances in mathematics
- ISSN:
- 0001-8708
- Page Range / eLocation ID:
- 108855
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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