### Abstract

Suppose f(x, y) is a binary form of degree d with coefficients in a field K ⊆ C. The K-rank of f is the smallest number of d-th powers of linear forms over K of which f is a K-linear combination. We prove that for d ≥ 5, there always exists a form of degree d with at least three different ranks over various fields. The K-rank of a form f (such as x^{3} y^{2}) may depend on whether -1 is a sum of two squares in K.

Original language | English (US) |
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Pages (from-to) | 5169-5177 |

Number of pages | 9 |

Journal | Proceedings of the American Mathematical Society |

Volume | 145 |

Issue number | 12 |

DOIs | |

State | Published - Dec 2017 |

### Keywords

- Binary forms
- Complex rank
- Real rank
- Stufe
- Sums of powers
- Sylvester
- Tensor decompositions

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

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## Cite this

Reznick, B., & Tokcan, N. (2017). Binary forms with three different relative ranks.

*Proceedings of the American Mathematical Society*,*145*(12), 5169-5177. https://doi.org/10.1090/proc/13666