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单词 MultiplicativelyIndependent
释义

multiplicatively independent


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

x1ν1x2ν2xnνn= 1

with  x1,x2,,xnX  and  ν1,ν2,,νn  implies that

ν1=ν2==νn= 0.

For example, the set of prime numbersMathworldPlanetmath is multiplicatively independent, by the fundamental theorem of arithmeticsMathworldPlanetmath.

Any algebraically independentMathworldPlanetmath set is also multiplicatively independent.

Evidently, {x1,x2,,xn} is multiplicativelyindependent if and only if the numbers logx1, logx2, …, logxn are linearly independentMathworldPlanetmath over .  Thus the Schanuel’s conjecture may be formulated as the

Conjecture.  If  {x1,x2,,xn}  is multiplicatively independent, then the transcendence degreeMathworldPlanetmath of the set

{x1,x2,,xn,logx1,logx2,,logxn}

is at least n.

References

  • 1 Diego Marques & Jonathan Sondow: Schanuel’s conjecture and algebraic powers zw and wz with z and w transcendental (2011). Available http://arxiv.org/pdf/1010.6216.pdfhere.
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更新时间:2025/5/4 5:08:10