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 $\log x_1$, $\log x_2$, …, $\log x_n$ are linearly independent over $\mathbb{Q}$.  Thus the Schanuel’s conjecture may be formulated as the

Conjecture.  If  $\{x_1,x_2,\mathrm{\dots },x_n\}$  is multiplicatively independent, then the transcendence degree of the set

$$\{x_1,x_2,\mathrm{\dots },x_n,\log x_1,\log x_2,\mathrm{\dots },\log x_n\}$$

is at least $n$.

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