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Multiplicative Independence

Multiplicative independence of nonzero complex numbers a_1,...,a_n, where n is a positive integer, means that the relation

a_1^(m_1)...a_n^(m_n)=1

forces m_1=...=m_n=0 whenever the exponents are integers (Richardson 2001). A set of nonzero complex numbers is multiplicatively independent iff every finite subset is multiplicatively independent. The empty set is multiplicatively independent.

For two integers a,b>1, multiplicative independence means that a^u!=b^v for all positive integers u and v. Equivalently, the ratio of their natural logarithms lna/lnb is irrational. For example, 2 and 3 are multiplicatively independent, whereas 2 and 4 are dependent because 2^2=4. Multiplicatively independent bases occur in the study of automatic sets (Albayrak and Bell 2023).

See also

Automatic Set, Base, Natural Logarithm

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References

Albayrak, S. and Bell, J. P. "Quantitative Estimates for the Size of an Intersection of Sparse Automatic Sets." Theoret. Comput. Sci. 977, 114144, 2023. https://doi.org/10.1016/j.tcs.2023.114144.Richardson, D. "Multiplicative Independence of Algebraic Numbers and Expressions." J. Pure Appl. Algebra 164, 231-245, 2001. https://doi.org/10.1016/S0022-4049(00)00156-0.

Cite this as:

Weisstein, Eric W. "Multiplicative Independence." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MultiplicativeIndependence.html

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