### Abstract

We consider the computation of two normal forms for matrices over the univariate polynomials: the Popov form and the Hermite form. For matrices which are square and nonsingular, deterministic algorithms with satisfactory cost bounds are known. Here, we present deterministic, fast algorithms for rectangular input matrices. The obtained cost bound for the Popov form matches the previous best known randomized algorithm, while the cost bound for the Hermite form improves on the previous best known ones by a factor which is at least the largest dimension of the input matrix.

Original language | English |
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Title of host publication | Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation |

Number of pages | 8 |

Publisher | Association for Computing Machinery |

Publication date | 2018 |

Pages | 295-302 |

ISBN (Print) | 978-1-4503-5550-6 |

DOIs | |

Publication status | Published - 2018 |

Event | 2018 ACM International Symposium on Symbolic and Algebraic Computation - New York, United States Duration: 16 Jul 2018 → 19 Jul 2018 |

### Conference

Conference | 2018 ACM International Symposium on Symbolic and Algebraic Computation |
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Country | United States |

City | New York |

Period | 16/07/2018 → 19/07/2018 |

### Keywords

- Polynomial matrix
- Reduced form
- Popov form
- Hermite form

## Cite this

Neiger, V., Rosenkilde, J. S. H., & Solomatov, G. A. (2018). Computing Popov and Hermite Forms of Rectangular Polynomial Matrices. In

*Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation*(pp. 295-302). Association for Computing Machinery. https://doi.org/10.1145/3208976.3208988