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Homemultiplicatively independent

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# multiplicatively independent

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

$x_{1}^{{\nu_{1}}}x_{2}^{{\nu_{2}}}\cdots x_{n}^{{\nu_{n}}}\;=\;1$ |

with $x_{1},\,x_{2},\,\ldots,\,x_{n}\in X$ and $\nu_{1},\,\nu_{2},\,\ldots,\,\nu_{n}\in\mathbb{Z}$ implies that

$\nu_{1}\;=\;\nu_{2}\;=\ldots=\;\nu_{n}\;=\;0.$ |

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

Conjecture. If $\{x_{1},\,x_{2},\,\ldots,\,x_{n}\}$ is multiplicatively independent, then the transcendence degree of the set

$\{x_{1},\,x_{2},\,\ldots,\,x_{n},\,\log x_{1},\,\log x_{2},\,\ldots,\,\log x_{% n}\}$ |

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 here.

## Mathematics Subject Classification

11J85*no label found*12F05

*no label found*

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