Boralevi, Ada; van Doornmalen, Jasper; Draisma, Jan; Hochstenbach, Michiel E.; Plestenjak, Bor
(2017).
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Uniform determinantal representations.
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SIAM journal on applied algebra and geometry, 1(1), pp. 415-441.
SIAM
10.1137/16M1085656

Text
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The problem of expressing a specific polynomial as the determinant of a square matrix of affine-linear forms arises from algebraic geometry, optimization, complexity theory, and scientific computing. Motivated by recent developments in this last area, we introduce the notion of a uniform determinantal representation, not of a single polynomial but rather of all polynomials in a given number of variables and of a given maximal degree. We derive a lower bound on the size of the matrix, and present a construction achieving that lower bound up to a constant factor as the number of variables is fixed and the degree grows. This construction marks an improvement upon a recent construction due to Plestenjak and Hochstenbach, and we investigate the performance of new representations in their root-finding technique for bivariate systems. Furthermore, we relate uniform determinantal representations to vector spaces of singular matrices, and we conclude with a number of future research directions.

## Item Type: |
Journal Article (Original Article) |
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## Division/Institute: |
08 Faculty of Science > Department of Mathematics and Statistics > Institute of Mathematics |

## UniBE Contributor: |
Draisma, Jan |

## Subjects: |
500 Science > 510 Mathematics |

## ISSN: |
2470-6566 |

## Publisher: |
SIAM |

## Language: |
English |

## Submitter: |
Olivier Bernard Mila |

## Date Deposited: |
17 Apr 2018 10:20 |

## Last Modified: |
17 Apr 2018 10:20 |

## Publisher DOI: |
10.1137/16M1085656 |

## BORIS DOI: |
10.7892/boris.109145 |

## URI: |
https://boris.unibe.ch/id/eprint/109145 |