## Abstract

We show that the spectral norm of a d-mode real or complex symmetric tensor in n variables can be computed by finding the fixed points of the corresponding polynomial map. For a generic complex symmetric tensor the number of fixed points is finite, and we give upper and lower bounds for the number of fixed points. For n = 2 we show that these fixed points are the roots of a corresponding univariate polynomial of degree at most (d-1)^{2} +1, except certain cases, which are completely analyzed. In particular, for n = 2 the spectral norm of d-symmetric tensor is polynomially computable in d with a given relative precision. For a fixedn > 2 we show that the spectral norm of a d-mode symmetric tensor is polynomially computable in d with a given relative precision with respect to the Hilbert-Schmidt norm of the tensor. These results show that the geometric measure of entanglement of d-mode symmetric qunits on C^{n} are polynomially computable for a fixed n.

Original language | English |
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Pages (from-to) | 2175-2215 |

Number of pages | 41 |

Journal | Mathematics of Computation |

Volume | 89 |

Issue number | 325 |

DOIs | |

Publication status | Published - 2020 |

Externally published | Yes |

## Keywords

- Anti-fixed and fixed points
- Computation of spectral norm
- D-mode symmetric qubits
- D-mode symmetric qunits on C
- Geometric measure of entanglement
- Homogeneous polynomials
- Spectral norm
- Symmetric tensors