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Hyperbolicity of the partition Jensen polynomials

  • Hannah Larson
  • Ian WagnerEmail author
Research
  • 30 Downloads

Abstract

Given an arithmetic function \(a: \mathbb {N}\rightarrow \mathbb {R}\), one can associate a naturally defined, doubly infinite family of Jensen polynomials. Recent work of Griffin et al. shows that for certain families of functions \(a: \mathbb {N}\rightarrow \mathbb {R}\), the associated Jensen polynomials are eventually hyperbolic (i.e., eventually all of their roots are real). This work proves Chen et al. conjecture that the partition Jensen polynomials are eventually hyperbolic as a special case. Here, we make this result explicit. Let N(d) be the minimal number such that for all \(n \ge N(d)\), the partition Jensen polynomial of degree d and shift n is hyperbolic. We prove that \(N(3)=94\), \(N(4)=206\), and \(N(5)=381\), and in general, that \(N(d) \le (3d)^{24d} (50d)^{3d^{2}}\).

Notes

Acknowledgements

The authors thank Ken Ono for suggesting this problem and providing advice. The authors are also grateful to Jesse Thorner for his help implementing Mathematica code that was used in the proof of Theorem 1.1. This research was supported by the National Science Foundation under Grant 1557960.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Emory UniversityAtlantaUSA

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