# multiplicatively independent

A set $X$ of nonzero complex numbers^{} is said to be multiplicatively independent iff every equation

$${x}_{1}^{{\nu}_{1}}{x}_{2}^{{\nu}_{2}}\mathrm{\cdots}{x}_{n}^{{\nu}_{n}}=\mathrm{\hspace{0.33em}1}$$ |

with ${x}_{1},{x}_{2},\mathrm{\dots},{x}_{n}\in X$ and ${\nu}_{1},{\nu}_{2},\mathrm{\dots},{\nu}_{n}\in \mathbb{Z}$ implies that

$${\nu}_{1}={\nu}_{2}=\mathrm{\dots}={\nu}_{n}=\mathrm{\hspace{0.33em}0}.$$ |

For example, the set of prime numbers^{} is multiplicatively independent, by the fundamental theorem of arithmetics^{}.

Any algebraically independent^{} set is also multiplicatively independent.

Evidently, $\{{x}_{1},{x}_{2},\mathrm{\dots},{x}_{n}\}$ is multiplicatively
independent if and only if the numbers $\mathrm{log}{x}_{1}$, $\mathrm{log}{x}_{2}$, …, $\mathrm{log}{x}_{n}$ are linearly independent^{} over $\mathbb{Q}$. Thus the Schanuel’s conjecture may be formulated as the

Conjecture. If $\{{x}_{1},{x}_{2},\mathrm{\dots},{x}_{n}\}$ is multiplicatively independent, then the transcendence degree^{} of the set

$$\{{x}_{1},{x}_{2},\mathrm{\dots},{x}_{n},\mathrm{log}{x}_{1},\mathrm{log}{x}_{2},\mathrm{\dots},\mathrm{log}{x}_{n}\}$$ |

is at least $n$.

## References

- 1 Diego Marques & Jonathan Sondow: Schanuel’s conjecture and algebraic powers ${z}^{w}$ and ${w}^{z}$ with $z$ and $w$ transcendental (2011). Available http://arxiv.org/pdf/1010.6216.pdfhere.

Title | multiplicatively independent |
---|---|

Canonical name | MultiplicativelyIndependent |

Date of creation | 2013-03-22 19:36:03 |

Last modified on | 2013-03-22 19:36:03 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 6 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 11J85 |

Classification | msc 12F05 |