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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.

Algebraic numbers can have three distinct conjugate elements alpha, beta, and gamma satisfying alphabeta=gamma (Drmota and Skałba 1995, p. 65). The relation alphabetagamma^(-1)=1 has integer exponents 1, 1, and -1, which are not all zero, so the triple is not multiplicatively independent.

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.Drmota, M. and Skałba, M. "Relations Between Polynomial Roots." Acta Arith. 71, 65-77, 1995. https://doi.org/10.4064/aa-71-1-65-77.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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