### Abstract

We investigate the number of real zeros of a univariate k-sparse polynomial f over the reals, when the coefficients of f come from independent standard normal distributions. Recently Bürgisser, Ergür and Tonelli-Cueto showed that the expected number of real zeros of f in such cases is bounded by [EQUATION]. In this work, we improve the bound to [EQUATION] and also show that this bound is tight by constructing a family of sparse support whose expected number of real zeros is lower bounded by [EQUATION]. Our main technique is an alternative formulation of the Kac integral by Edelman-Kostlan which allows us to bound the expected number of zeros of f in terms of the expected number of zeros of polynomials of lower sparsity. Using our technique, we also recover the O (log n) bound on the expected number of real zeros of a dense polynomial of degree n with coefficients coming from independent standard normal distributions.

Original language | English |
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Title of host publication | ISSAC 2020 - Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation |

Editors | Angelos Mantzaflaris |

Publisher | ACM |

Pages | 273-280 |

Number of pages | 8 |

ISBN (Electronic) | 9781450371001 |

DOIs | |

Publication status | Published - 21 Jul 2020 |

MoE publication type | A4 Article in a conference publication |

Event | International Symposium on Symbolic and Algebraic Computation - Virtual, Kalamata, Greece Duration: 20 Jul 2020 → 23 Jul 2020 Conference number: 45 https://issac-conference.org/2020/program.php |

### Conference

Conference | International Symposium on Symbolic and Algebraic Computation |
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Abbreviated title | ISSAC |

Country | Greece |

City | Kalamata |

Period | 20/07/2020 → 23/07/2020 |

Internet address |

### Keywords

- random polynomials
- real tau conjecture
- sparse polynomials

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## Cite this

Jindal, G., Pandey, A., Shukla, H., & Zisopoulos, C. (2020). How Many Zeros of a Random Sparse Polynomial Are Real? In A. Mantzaflaris (Ed.),

*ISSAC 2020 - Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation*(pp. 273-280). ACM. https://doi.org/10.1145/3373207.3404031